The last chapter of Mathematical Structures was about how to spot false proofs. Of course, I am not the first to do this. A curious chain (I may tell about this later) led me to *The Sophismata of Richard Kilvington*.

Richard Kilvington (ca.1302–1361) was a Franciscan who worked in Oxford for a time in the early 14th century. He was associated with the Merton mathematicians, though it seems that he was not actually at Merton himself (despite some later reports).

He worked in terminist logic, mathematical physics, and the new theology. In particular, he refuted Aristotle’s claim that *metabasis* (using arguments or methods from one branch of science in a different branch) was illicit.

On infinity, he says that integers are potentially infinite because one can always find a larger integer, but not actually infinite since there is no single infinite number.

His best-known work is his *Sophismata*, a discussion of paradoxes or questionable truths. These have been translated, edited and commented by Norman Kretzmann and Barbara Ensign Kretzmann, *The Sophismata of Richard Kilvington*, Cambridge University Press, 1990. (But beware; there is another book with the same title and editors, published by Oxford University Press in the same year, which is a collation and transcription of the existing Latin texts; unless you are expert in mediaeval Latin, this may not be what you want.)

Here, courtesy of Google Books, is the list of sophismata which Kilvington discusses. I think they throw an interesting light on the philosophy of the time. Traditionally ridiculed as being about angels dancing on pinheads, the subject was really much more. Simply reading this list, you will spot connections with Zeno’s paradox, infinity, infinite divisibility, change, the basis of empirical knowledge, and one of the obsessions of the Merton mathematicians, quantification. These are things which still perturb philosophers and others. Don’t be put off by all the stuff about whiteness; this is just an arbitrary attribute which can change and might be quantifiable.

- Socrates is whiter than Plato begins to be white.
- Socrates is infinitely whiter than Plato begins to be white.
- Socrates begins to be whiter than Plato begins to be white.
- Socrates begins to be whiter than he himself begins to be white.
- Socrates will begin to be as white as he himself will be white.
- Socrates will begin to be as white as Plato will be white.
- Socrates will be whiter than Plato will be white in any of these.
- Socrates will be precisely as Plato will be white in any of these.
- Socrates will be as white as Plato will cease to be white.
- Socrates will be two times whiter than Plato will be white at instant A.
- Something has produced degree B of whiteness.
- Socrates has traversed distance A.
- Socrates will traverse distance A.
- Socrates will begin to traverse distance A, and Socrates will begin to have traversed distance A, and he will not begin to traverse distance A before he will begin to have traversed distance A.
- Distance A begins to have been traversed.
- A begins to be true.
- A and B will be true.
- A was moving continuously during some time after B, and A is not moving.
- Socrates will as quickly cease to move as he will move.
- Socrates will as quickly have been destroyed as he will have been generated.
- A bgins to intensify whiteness in some part of B, and each proportional part in B will without interval be diminished.
- A begins to whiten some part in B, and no part in B will be whiter than it now is white.
- A will generate whiteness up to point C, and no whiteness will be immediate to point C.
- D will begin at the same time to have been divided and not divided.
- A will begin to have been divided from B.
- A will begin to be per se whiter than B.
- Socrates will begin to be able to traverse distance A.
- Distance A will begin to have been traversed by Socrates.
- Socrates will move over some distance when he will not have the power to move over that distance.
- Socrates moves two times faster than Plato.
- Socrates and Plato will begin to move equally fast.
- Socrates does not move faster than Plato.
- Socrates will move faster than Socrates now moves.
- Plato can move uniformly during some time and as fast as Socrates now moves.
- Socrates will begin to be able to move at degree A of speed.
- Socrates will begin to be able to move stone A.
- Socrates can as quickly have the power to move stone A as Plato will have the power to traverse distance C.
- Plato can begin to be the strongest of the men who are in here.
- A and half of A begin at the same time to be destroyed by agent B.
- It is infinitely easier to make C be true than to make D be true.
- B will make C true.
- It is infinitely easier for B to make it be the case that the proposition “Infinitely many parts of A have been traversed” is true than to make it the case that the proposition “All of A has been traversed” is true.
- Infinitely sooner will A be true than B will be true.
- As many proportional parts in A Socrates will traverse as Plato.
- You know this to be everything that is this.
- You know this to be Socrates.
- You know that the King is seated.
- A is known by you.

A link or ISBN for the more useful tome would be helpful đź™‚

http://www.cambridge.org/gb/knowledge/isbn/item5736934/?site_locale=en_GB

On the one hand Aristotle gives us the logic of analogy (παραδειγμα). On the other hand he cautions us that different paradigms may have no common measure. It seems these Immortals are always getting ahead of their timeâť˘

How much drama in Plato’s Heaven when Heraclitus and Parmenides are reconciledâť˘

• Differential Logic and Dynamic Systems

Or would it be but anticlimbatic that Sisyphus gradually wears down the mountain?

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Dealing with qualitative change in logical terms has long been of interest to me. It became something of a hot topic in Artificial Intelligence research back in the 1980s — work by Ben Kuipers and Ken Forbus especially comes to mind. Many of the settings where I worked during that decade required me to find bridges between qualitative (logical) and quantitative (statistical) research methods. I recall describing my efforts in that vein to one of my Master’s thesis advisers as

commuting the qualitative theory of differential equations into the differential theory of qualitative equations. Another way to say it would beexchanging a change of quality for a quality of change. Here’s another piece I wrote in that line:• Differential Propositional Calculus