The general question of this post is of the relation between
The specific question is why Pascal juxtaposes Abraham and Gideon in two fragments of the Pensées.
A simple answer to the specific question is that
Caravaggio, Sacrifice of Isaac, 1603, Uffizi
By the account of Martha Nussbaum, philosophy is one of two things:
Nussbaum prefers the first, though having appeared in a film that promotes the second.
I watched and enjoyed the film, which is by Astra Taylor and is called Examined Life (2008). I first found it through a touching fragment, featuring a stroll in San Francisco by Judith Butler and Taylor’s sister Sunaura. Because they have a conversation at all, and on the theme that we all need one another’s help, the film becomes less subject to Nussbaum’s charge:
Portraying philosophers as authority figures is a baneful inversion of the entire Socratic process, which aimed to replace authority with reason.
In some of the Pensées, Pascal contrasts reason with instinct, passions, folly, the senses, and imagination.
Here I investigate Pascal’s raison, after one session of an ongoing discussion of the Pensées that is being carried out on Zoom.
To this post, I am adding this introduction in July 2021. I have returned to some of the ideas of the post, and I see that I left them in a jumble. They may still be that, but I am trying to straighten up a bit.
Beyond this introduction, the post has three parts. Part III takes up more than half of the whole post and consists of my notes on
In Anderson’s article I see – as I note below – ideas that are familiar, thanks to my previous reading of philosophers such as Robin George Collingwood, Mary Midgley, and Robert Pirsig. Henry David Thoreau may not exactly be one of those philosophers, but he is somehow why I came to write this post in the first place.
Here is a table of contents for the whole post:
Pacifism is properly pacificism, the making of peace: not a belief or an attitude, but a practice. Mathematics then is pacifist, because learning it means learning that you cannot fight your way to the truth. Might does not make right. If others are going to agree with you, they will have to do it freely. Moreover, you cannot rest until they do agree with you, if you’ve got a piece of mathematics that you think is right; for you could be wrong, if others don’t agree.
This article gathers, and in some cases quotes and examines, popular articles about R. G. Collingwood (1889–1943).
By articles, I mean not blog posts like mine and others’, but essays by professionals in publications that have editors.
By popular, I mean written not for other professionals, but for the laity.
Large parts of this post are taken up with two subjects:
The notion (due to Collingwood) of criteriological sciences, logic being one of them.
Gödel’s theorems of completeness and incompleteness, as examples of results in the science of logic.
Like the most recent in the current spate of mathematics posts, the present one has arisen from material originally drafted for the first post in this series.
In that post, I defined mathematics as the science whose findings are proved by deduction. This definition does not say what mathematics is about. We can say however what logic is about: it is about mathematics quâ deduction, and more generally about reasoning as such. This makes logic a criteriological science, because logic seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are, in a crude sense to be refined below, that
everything true in all possible mathematical worlds can be deduced;
some things true in the world of numbers can never be deduced;
the latter theorem is one of those things.
I say that mathematics is the deductive science; and yet there would seem to be mathematicians who disagree. I take up two cases here.
Mathematics “has no generally accepted definition,” according to Wikipedia on September 15, 2020, with two references. On September 14, 2023, the assertion is, “There is no general consensus among mathematicians about a common definition for their academic discipline”; this time, there are no references.
I suggest that what really has no generally accepted definition is the subject of mathematics: the object of study, what mathematics is about. Mathematics itself can be defined by its method. As Wikipedia says also (as of either date given above),
it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
I would put it more simply. Mathematics is the science whose findings are proved by deduction.
The right triangle whose legs are 3 and 4 has hypotenuse 5, because the square on it is
(4 − 3)2 + 2 ⋅ (4 ⋅ 3),
which is indeed 25 or 52. This is also
42 + 32.
The bulk of this post is a summary of the chapter on art in Collingwood’s Speculum Mentis: or The Map of Knowledge (1924). The motto of the book is the first clause of I Corinthians 13:12:
Βλέπομεν γὰρ ἄρτι δι’ ἐσόπτρου ἐν αἰνίγματι
For now we see through a glass, darkly
The chapter “Art” has eight sections:
Polytropy 