Since our walk through Summa Contra Gentiles is going so well, why not let’s do the same with Pascal’s sketchbook on what we can now call Thinking Thursdays. We’ll use the Dutton Edition, freely available at Project Gutenberg. (I’m removing that edition’s footnotes.)
Update Comments fixed.
The difference between the mathematical and the intuitive mind1.—In the one the principles are palpable, but removed from ordinary use; so that for want of habit it is difficult to turn one’s mind in that direction: but if one turns it thither ever so little, one sees the principles fully, and one must have a quite inaccurate mind who reasons wrongly from principles so plain that it is almost impossible they should escape notice.
But in the intuitive mind the principles are found in common use, and are before the eyes of everybody. One has only to look, and no effort is necessary; it is only a question of good eyesight, but it must be good, for the principles are so subtle and so numerous, that it is almost impossible but that some escape notice. Now the omission of one principle leads to error; thus one must have very clear sight to see all the principles, and in the next place an accurate mind not to draw false deductions from known principles.
All mathematicians would then be intuitive if they had clear sight, for they do not reason incorrectly from principles known to them; and intuitive minds would be mathematical if they could turn their eyes to the principles of mathematics to which they are unused.2
The reason, therefore, that some intuitive minds are not mathematical is that they cannot at all turn their attention to the principles of mathematics. But the reason that mathematicians are not intuitive is that they do not see what is before them, and that, accustomed to the exact and plain principles of mathematics, and not reasoning till they have well inspected and arranged their principles, they are lost in matters of intuition where the principles do not allow of such arrangement. They are scarcely seen; they are felt rather than seen; there is the greatest difficulty in making them felt by those[Pg 2] who do not of themselves perceive them. These principles are so fine and so numerous that a very delicate and very clear sense is needed to perceive them, and to judge rightly and justly when they are perceived, without for the most part being able to demonstrate them in order as in mathematics; because the principles are not known to us in the same way, and because it would be an endless matter to undertake it. We must see the matter at once, at one glance, and not by a process of reasoning, at least to a certain degree. And thus it is rare that mathematicians are intuitive, and that men of intuition are mathematicians, because mathematicians wish to treat matters of intuition mathematically, and make themselves ridiculous, wishing to begin with definitions and then with axioms, which is not the way to proceed in this kind of reasoning. Not that the mind does not do so, but it does it tacitly, naturally, and without technical rules; for the expression of it is beyond all men, and only a few can feel it.3
Intuitive minds, on the contrary, being thus accustomed to judge at a single glance, are so astonished when they are presented with propositions of which they understand nothing, and the way to which is through definitions and axioms so sterile, and which they are not accustomed to see thus in detail, that they are repelled and disheartened.
But dull minds are never either intuitive or mathematical.
Mathematicians who are only mathematicians have exact minds, provided all things are explained to them by means of definitions and axioms; otherwise they are inaccurate and insufferable, for they are only right when the principles are quite clear.
And men of intuition who are only intuitive cannot have the patience to reach to first principles of things speculative and conceptual, which they have never seen in the world, and which are altogether out of the common.4
1From Allan Bloom The Closing of the American Mind: How Higher Education Has Failed Democracy and Impoverished the Souls of Today’s Students (p. 52):
Every Frenchman is born, or at least early on becomes, Cartesian [the mathematician above] or Pascalian [the intuitive]…Descartes and Pascal represent a choice between reason and revelation, science and piety, the choice from which everything else follows…These great opponents whom no snythesis can unite—the opposition between bon sens and faith against all odds—set in motion a dualism…
It was, therefore, very French of Toucqueville to say that the Americans’ method of thought was Cartesian…
2The great fallacy is to suppose we can do with only one of these types (even inside one body). American and British thought plunges headlong into the mathematical—we are all Cartesians here. This isn’t a new observation. Tocqueville said “each American appeals to the individual exercise of his own understanding alone. America is therefore one of the countries in the world where philosophy is least studied, and where the precepts of Descartes are best applied…they follow his maxims because this very social condition naturally disposes their understanding to adopt them.”
Strict Cartesianism leads to scientism and the worship of rationality and reason as if these could live without intellection, what Pascal called intuition. No mathematician could even begin to think without intellection. Intuition, used in this special sense, is necessary and prior to logic, mathematics, and ratio. Axioms, for instance, are not provided by rationality. Pure rationality is always incomplete. I’ll have much more to say about this in the coming weeks.
3It is well to put it here the fallacy that says that because sometimes our intuitions fail us that they always do. Sometimes our mathematical reason also fails us, but nobody would claim that therefore all of mathematics should be tossed or is suspect (except radical skeptics; paradoxically, personages only found in Western universities).
4Relying only on one leads to rank pedantry, sterility, and blind alleys.