[hist-analytic] "Ordine geometrico demonstrata"

Jlsperanza at aol.com Jlsperanza at aol.com
Mon Feb 23 21:12:02 EST 2009

I'm enjoying the exchange Bishop/Bayne. I'd like to drop to the annals of  

Spinoza's "more geometrico"

This seems to have been more  influential on the Continent than on the Isles, 
but then S. N. Hampshire did  write a Penguin book on Spinoza.

I do not know as much as I'd need to  know about Greek mathematics, but I 
wonder sometimes if geometry is more basic  than arithmetics. Or why would 
Spinoza dub his thing 'more geometrico' rather  than 'more arithmetico'?

>From online source:

"Baruch de Spinoza  attempted to improve Descartes philosophy 
by mathematically axiomatize  philosophical thought into a coherent system 
(starting definitions, axioms,  postulates, theorems with their demonstrations) 
= more geometrico (in the  geometrical manner)."

I note that his fist book was on the principles of philosophy of Descartes  
'MORE' geometrico demonstratae.
The 'Ethica ORDINE geometrico demonstrata' is later.
I always found that the way Grice 'caricatures' the 'formalists' (as he  then 
called them) in 'Logic and Conversation' (WOW, ii) -- first page -- is  right 
into the 'heritage of Spinoza' if you think about it (and even if you  don't).
>From Bishop's notes:
I note that in his notes Bishop mentions the "Prolegomena" remarks by Grice  
(re: Strawson (1952):

>"expressions which are candidates for being natural analogues to  logical 
By way of critique, as Bishop charmingly puts it: 
>I was rather pleased to see Grice's list, 
     [which includes Strawson 1952 -- and hence the  title "LOGIC and 
Conversation" for the second lecture]
>for it contains many examples which I have come 
>across and which I have previously myself felt to be unsound. 
Exactly. But it would have been _anathema_ to contradict Strawson where I  
come from!
>Pleased to see them, I suppose, because my familiarity 
>with the literature has been insufficient for me to have seen 
>these criticised before. 
And Grice is being charming himself. For Strawson does acknowledge "Mr. H.  
P. Grice" as his logic tutor "from whom I have never ceased to learn from logic 
 since he taught me in that area" (or words to that effect). A bit like  
overqualifying the thing. ("just a logic tutor") but let that pass. (The memoirs  
by J. D. Mabbott here are a delight. The man was Scots born, Mabbott, and in 
his  "Oxford memoirs" (published by a minor publisher of Oxford) wants to say 
that  Strawson was by far his most brilliant student (tutors have to be careful 
 there). He goes on to add, by way of support, "And so seems to have thought, 
 too, his other tutor at St. John's, the 'excellent' H. P. Grice" -- or words 
to  that effect. Or 'intelligent'. Mabbot retells in detail how Strawson came 
out,  poor thing, with a second, rather than a first. -- But we don't have to 
spread  _that_ news.

But this notes how parochial the whole thing was for we have:
1940s. Seminars Grice/Strawson (indeed Strawson was appointed Grice's tutee  
-- _before_ the war, as I recall).
1952. Strawson, Introduction to logical theory. Methuen (crediting "H. P.  
Grice" -- where? In connection with 'informativeness' -- one brief  footnote).
1966. Chomsky cites "A. P. Grice" in Aspects of the theory of syntax" with  
regard to the logical behaviour of 'and' (implicating: 'and then').
1967. Logic and Conversation, by Grice. Resuming the polemic.
------ But by 1967, Grice was able to provide a 'larger' picture. Strawson  
comes out as an 'informalist' (later a 'neo-traditionalist'), opposing the  
'doctrine' of the formalists (Quine, Russell/Whitehead) later called  
'modernists' (heirs of PM). 
Grice's point is that both camps _share_ the view that there _is_ a  
divergence between, to use Bishop's wording -- citing Grice --:
(a) "expressions which are candidates for being natural analogues to  logical 
(b) "logical constants proper". Grice mentions the truth-functors but also,  
let's recall the universal quantifier, the existential quantifier, and the 
iota  operator -- which are not truth-functors. Hence his label of them as 
'(formal)  [i.e. structural] devices'. 
----- Personally, other than the Grice/Strawson polemic, I don't think I  
recall anything as systematic in the American literature. Grice does suggest  
that some philosophers (before him) have noted that there may be something  
'unsound' about this -- i.e. that the divergence is _seeming_ rather than  real.
(Also, Grice pedantically but so rightly puts it, "the thesis that there  is, 
or there _seems_ to be a divergence between..." -- for as things develop, he  
would claim the divergence is a matter of implicature, i.e. _seeming_ and not 
 truth-conditional, in a way -- cfr. his "Valediction" or retrospective 
epilogue  which _notably_ focuses on the polemic with Strawson rather than any big 
more  fashionable issue of implicature per se -- I found that refreshing, that 
he goes  back to the source of inspiration of it all: Strawson and his 
Now, I once tried -- and I think I did list in my PhD dissertation -- all  
the postulates Grice _requires_ for the 'formalist' credo. They are quite a  
bunch. They all sound, now having refreshed my Greek Mathematics (with Thomas,  
and his two-volume Loeb) that it's the _axiomatic_ treatment, the Spinoza  
'ordine geometrico' that he is talking about. Gentzen was just publishing those  
things (right?) and Grice will have occasion to present a cruder version of the 
 'formalist' credo in his contribution to the festschrift for Quine ("Vacuous 
 Name", in Words and Objections, ed. Davidson/Hintikka, 1969).
In the Quine festschrift, Grice does stick with Gentzen's idea that for  each 
operator (or device) we do need a (+) rule and a (~) rule: introduction and  
elimination, and proceeds to provide truth-table compatible introductions for  
the functors and the quantifiers. This is clearer than most of what Grice 
really  does in "Logic and Conversation" which is merely jocular in the best 
sense of  'jocular'. 
I think it was Aloysius Martinich (the Russian-born philosopher at  
UTexas/Austin) who noted this: they say 'Logic and Conversation' (WOW, ii) is a  
_masterpiece_ of an essay, but on second thoughts, it's pretty inconclusive and  
there seems to be a quick change of topic. 
When Bishop summarises the chapter in his online notes, he goes straight to  
Grice's definition of his term of 'art', "implicature" (I noted that Sidonius  
uses 'implicatura' in his Loeb -- cited by Short/Lewis, though!). But it 
should  be noted that the discussion of 'implicature' (a special section) 
_follows_ the  account of the 'problem' (as Grice saw it) and the two tenets who share 
a  'common mistake'. 
I would think that Grice believed to his last day that the formal devices  
can indeed be _saved_ by all means. I.e. an axiomatic treatment is possibly the  
best thing to do cognitively. 
But he was ready to admit or allow that in other areas, notably, the  'ordine 
geometrico' (which seems to work best with 'extensional' rather than  
'intensional' categories, including 'modal') seems more of a strain (if that's  the 
word) than anything 'liberatory'.


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