Pascal’s Pensées, A Tour: II

PascalSince 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.)

Previous post.


There are different kinds of right understanding; some have right understanding in a certain order of things, and not in others, where they go astray. Some draw conclusions well from a few premises, and this displays an acute judgment.

Others draw conclusions well where there are many premises.

For example, the former easily learn hydrostatics, where the premises are few, but the conclusions are so fine that only the greatest acuteness can reach them.

And in spite of that these persons would perhaps not be great mathematicians, because mathematics contain a great number of premises, and there is perhaps a kind of intellect that can search with ease a few premises to the bottom, and cannot in the least penetrate those matters in which there are many premises.

There are then two kinds of intellect: the one able to penetrate acutely and deeply into the conclusions of given premises, and this is the precise intellect; the other able to comprehend a great number of premises without confusing them, and this is the mathematical intellect. The one has force and exactness, the other comprehension. Now the one quality can exist without the other; the intellect can be strong and narrow, and can also be comprehensive and weak.1


Those who are accustomed to judge by feeling do not understand the process of reasoning, for they would understand at first sight, and are not used to seek for principles. And others, on the contrary, who are accustomed to reason from principles, do not at all understand matters of feeling, seeking principles, and being unable to see at a glance.2


1A tacit premise here are the levels of intellectual ability. Einstein was famously considered by his peers to be less than an immaculate mathematician. But Einstein’s mathematical skills, relatively (get it? get it?) poor as they were, were orders of magnitude greater than, say, those of the homme moyen. I think what Pascal means is that in any of us one form of thinking dominates the other, and perhaps this isn’t by choice: he cannot be saying that each of us is either a great mathematician or a terrific inductionist.

The two manners of thinking reminds of the little science fiction I remember. Asimov’s men versus his robots, which he always had men call “logical, not reasonable.” Many see this as a goal. Peter Kreeft warns of the consequence: (in Socratic Logic, p. 35) “a new species of human mind has appeared: one that does not know the difference between a human mind and a computer”. Who would aspire to be a walking calculator?

2I prefer induction over intuition, inductive over intuitive, and so forth. Our culture with just suspicion looks down on “intuitionist” modes of thinking, for this is where “feelings” reign and charlatans of every stripe live. Robotics is promoted, philosophy maligned, or rather it is aligned to robotics. But our “feelings” are not Pascal’s “feelings.” Because we don’t keep this straight, we incorrectly condemn, or rather fail to reward, inductivist thinking. Deepak Chopra in an intuitionist; Einstein was an inductivist.

What Pascal shows is that there is a touch of truth to the academic “different ways of knowing” fraud. Some of us come at the truth at once, and some by formal rigorous calculation. Surely the path taken does not matter if one arrives at the proper destination?

Except the difficultly is that not all truths can be reached by calculation. Rigorous calculation proves this! Right, Gödel? Empiricism is circular and all mathematics, logic, and rationality must begin on inductivist grounds. Inductivist thinking is thus superior because it holds and discovers greater truths.

Quoting in Groarke p. 293): “‘Not to know of what things one may demand a demonstration, and of what things one may not’ is, for Aristotle, to lack education.” And (pp. 298-299): “If modern philosophers have generally focused on logic and deduction as the most authoritative source of knowledge, Thomas [Aquinas], following after Plotinous, considers intellectus or “understanding” to be a higher form of knowing.”

Intellection, or inductivism, is to know “through a species of immediate, instantaneous illumination.” It is to see the universal in the particular, to see the essence in an example. It is a gift.

On Thomas’ account, what we call reasoning [ratio] is, in fact, an inferior form of knowing. We do not immediately grasp the implications of the facts that we know. We are not intelligent enough for that! Because we lack intelligence, we must reason through a middle term [as in syllogisms]. Discovering the truth requires effort. We need an aid, a crutch. This is what logic provides. Because “of the dimness of the intellectual light in [our] souls,” we must reason things out, moving step by step from premise, to premise, to conclusion. We see part of the truth and use it to logically calculate another part of it. Although Thomas could be said to devote his whole theological and philosophical career to logical argument, he himself recognized that this form on insight is inferior.

Warning Any commentary on global warming in this post will be deleted.


  1. DAV

    There has been a surfeit of applesauce lately.

    I don’t think there is much difference between intuition and logic. It just isn’t formal.

  2. View from the Solent

    I’m not so sure, DAV.
    Often, the intuition (aka insight?) comes first. Then followed by the rigorous logic to prove it. That latter step might not happen for a long time (Fermat’s Last), if ever (Riemann hypothesis?).

  3. Sander van der Wal

    Asimov’s robots were almost completely unable to formulate new hypotheses, reason about them, test them, reject them, or occasionally start using them as new theories. They were stuck with the Three Laws of Robotics. The best they managed was to generalize those laws from protecting a human to protecting humanity, and as a result exterminating all other intelligent life in the Milky Way.

  4. Thanks for posting this Matt, and particularly for a post that will not invite AGW sermons on AGW from our local evangelical warmist.

  5. DAV


    I believe it may depend on definition. Intuition is subconscious but not necessarily illogical. You could think of it as proposal.


    Not all, though. There was The Bicentennial Man. Some of the 3-Law conflicts were resolved in novel ways IIRC. To do that seems to require formulation and testing of hypotheses.

    Our goals may direct our thinking more than we imagine or care to believe.

  6. (I decided to post this comment separately in case you decided to delete my thank you with its reference to posting on AGW.)

    Anybody read Isaiah Berlin’s famous essay “The Hedgehog and the Fox”?
    The title refers to an old Greek aphorism “πόλλ’ οἶδ’ ἀλώπηξ, ἀλλ’ ἐχῖνος ἓν μέγα (the fox knows many things, but the hedgehog knows one big thing)”.
    Organic chemistry is a subject for those who can know many things–I almost flunked out of Caltech for conditionals in that subject when I was still a chemist. And it’s also remarkable that organic chemists do really well in computer programming
    When I was teaching thermo to pre-med students I told them remember the two laws of thermo and the definitions of thermodynamic functions and derive everything from that–they never did that, always wanted a specific equation for a specific application. Are organic chemistry, computer programming and medicine fox things and philosophy and theology the hedgehog things?

  7. Solent, with regard to intuition the best example I can think of (putting on my chemist hat) is that of Kekule who dreamed of the Ouruboros (sp?), the snake that eats its own tail, and thence the benzene ring.

  8. Sander van der Wal

    “Inductivist thinking is thus superior because it holds and discovers greater truths.”

    No, that doesn’t follow. All true statements are each as true as all other true statements. Inductivist thinking might result in more interesting true statements. If you can find them among all the half-truths and untruths you will think of.

  9. Briggs


    Your ratio has let you down. It does follow. True statements are true, but how you know they are true is not the same. Intellection/induction lead you to truths inaccessible by ratio/rationality, and also to truths accessible by those means. Therefore it is a superior form of thinking.

  10. Gary

    “A general “law of least effort” applies to cognitive as well as physical
    exertion. The law asserts that if there are several ways of achieving the
    same goal, people will eventually gravitate to the least demanding course
    of action. In the economy of action, effort is a cost, and the acquisition of
    skill is driven by the balance of benefits and costs. Laziness is built deep into our nature.”

    ― Daniel Kahneman, in his book, Thinking, Fast and Slow, explores the two thought systems identified above by Pascal. Each has advantages and disadvantages depending on the circumstances. It makes complete sense that one becomes the dominant pattern of thought.,_Fast_and_Slow for a synopsis. for a video lecture.

  11. Nate

    I like your distinction between intuition and induction.

    Psychologists have the MBTI test, which assigns a person various “types” based on their answers to a series of questions. Leave aside for the moment how much a questionnaire really reveals – I tend to see it as an easy way to categorize people, in general, based on how they seem to make their decisions and carry out the business of their work and life.

    There’s “Intutive” vs “Sensing” personality types, which kind of describes Pascal’s separation above. But there’s also “Feeling” vs “Thinking” (and “Judging/Perceiving” and “Introversion/Extroversion”).

    Perhaps we could classify Deepak Chopra, the intuitionist, as an “Intuitive Feeling” person. Einstein, the inductivist, was an “Intuitive Thinking” type of person?

    It’s a surprisingly robust classification mechanism, providing you leave it in generalities and recognize that everyone is a little blend of everything, with some things stronger than others (as you said).

  12. Sander van der Wal


    So? You axiomatize a bit and then you can prove that your inductive theory follows from the axioms. And they you compare it to reality, the great revealer of false hypotheses. If you’re lucky, it is proven false.

  13. Briggs


    To “axiomatize a bit” it to think inductively. A bit.

  14. Like DAV, I’m wondering if there really are two separate ways of reaching a conclusion. View from the Solent says: “Often, the intuition (aka insight?) comes first. Then followed by the rigorous logic to prove it.” Perhaps the intuition comes first because some people go through the logic subconsciously. Once the conclusion has been reached, then there is an effort to find the logic/reasoning behind it. It’s an unprovable hypothesis, I know, but to me it makes sense that intuition is just subconscious logic at work. (One could say one is formal, the other not as did DAV.)

  15. Briggs


    There are sometimes two ways, but not always. “Axiomatizing a bit”, as Sander put it, cannot be done mathematically. It can only be done inductively.

  16. As I said in a previous comment, much of this discussion has been set forth by Michael Polanyi…Google Personal Knowledge, Tacit Knowledge.

  17. brad tittle

    There was an article posted somewhere else suggesting that thyroid cancer detection has done nothing to help thyroid cancer healing.

    Other places there are similar tales. Our host has suggested that the ideal breast cancer tool is to just say NO. He has quantized the explanation, but even so there is just a hint of induction. The numbers are always death. But the living have the answer. They live. That is the truth. They live with maladies. The live with deformities. They live with addictions. They defy death with every step. But the analysis continues to be of death because we can find death. It would appear easy to find life, but life doesn’t enumerate as well as we hope. The premises multiply too fast. Deaths have nearly as many premises, but the unseen premises are as unseen as in the living.

    Ask Coca-Cola about their foray into science. The new flavor they picked was scientifically picked. Test groups were formed and evaluated all the flavors. They made a better tasting product according to the test groups. Out the door went the new product and the failure was epic. They learned something from their mistake. Never question the taste of the customer. Don’t tell the customer what they like. Give them something to like. Market those somethings so that they know they are there. Keep moving forward. Want a new flavor profile? Make a new product.

    There are things to embrace strongly. Dry tinder makes fires start faster. There are things to embrace lightly. Checking for cancer…

    I wonder if I have said anything.

  18. Scotian

    I’ve always liked this quote:
    “It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”
    By Alfred North Whitehead.
    I think that this is a better distinction to the one made in today’s post.

    From above “For example, the former easily learn hydrostatics…” This does not seem like a good example. Does he mean hydrodynamics or is one too simple and the other too difficult? Pascal’s law anyone?

    Also “There are then two kinds of intellect: the one able to penetrate acutely and deeply into the conclusions of given premises, and this is the precise intellect; the other able to comprehend a great number of premises without confusing them, and this is the mathematical intellect.” I would have guessed that the former was the mathematical intellect and the latter more in line with physics but than Pascal lived before the separation of the disciplines. I have noticed in my career that it is easier for physics students to do well in upper year math courses than for math students to do well in upper year physics courses. Does this say something about modern math education?

    Pascal was also a Jansenist. You can’t like that Briggs.

  19. Scotian: I’m not sure, but my guess would be that your last question says more about physics and how it is different from straight math. Physics seems to require more abstract thinking. At least that was my experience.

  20. Sheri–I disagree that physics requires more abstract thinking. It does require the ability to go from an abstract law to a particular situation–for example, to go from notions of symmetry to finding the symmetry adapted coordinates for the harmonic oscillator problem of CO2 (pace Briggs–no AGW). In math you need the the ability to remember all the various theorems, corollaries, etc., besides the ability to string them together logically–they’re abstract in the extreme. I don’t know what rings and Banach spaces are because I can’t relate them to something concrete.
    Scotian, I’ll agree that physics students do better in upper level math courses that can be connected to real stuff–e.g. group theory, boundary value problems, calculus of variations, linear algebra. I think math students have a problem in extending a fundamental law or principle to a particular situation, i.e. connecting it to reality.

  21. At the risk of being a pedant, Asimov’s robots were none of the things described by Sander. The Bicentennial Man being a case in point. (And if you have not read the book, the film version with Robin Williams is close enough.)

    The point of the Three Laws was not to restrict the intellectual capabilities of the robots, but to explore how they might be navigated around in various ways (“law of unintended consequences”) , while still remaining true to those laws. Asimov’s robots did not exterminate all other intelligent life. Although I can’t speak for other writers who may have played in Asimov’s universes, as I haven’t read those non Asimov books.

    One critical aspect of intellect that seems left out of this discussion (so far) is the role of experience. Or what one might call ‘wisdom.’ There is a reason why old people are considered ‘wise’ but very few teenagers earn the label. The more experience one has, the more capable one is to re orientate problems into different moulds, one of which may lead to a solution.

  22. MattS

    The biggest problem with Asimov’s three laws, is what happens when the robots figure out that the biggest threat to humanity is humanity itself.

    See I-Robot.

  23. Sander van der Wal


    But whether the axiomatization produces true statements or not is not inductive. It is the comparison to reality that does it. Which makes the dichotomy between deductive and inductive reasoning a false one.

    Make up a new inductive hypothesis based on experience and earlier hypotheses. Derive deductively the logical consequences. Compare these to reality. Winnow the bits that reality doesn’t like. Repeat.

  24. @Matt

    “The biggest problem with Asimov’s three laws, is what happens when the robots figure out that the biggest threat to humanity is humanity itself.”

    I don’t think Asimov ever explored that theme – the argumentum ad absurdum – of his premise. If so, what story please? Otherwise, it was explored in the classic short story, With Folded Hands, by Jack Williamson. Probably still my favourite all time short SF story.


    What’s the relevance of the deduction/induction distinction to the nature of teaching reasoning? Deduction/induction are both concerned with establishing universal laws. Which strikes me as primarily an epistemological concern of little relevance to intelligence or practical problem solving.

  25. Sander van der Wal

    @Will Nitschke

    That theme was explored by other SF authors working in the Asimov Robot Universe. People like Benford and Brin. No book titles, it has been a while and the books came from the library.

    Regarding the relevance to the nature of teaching reasoning, I have no particular opinion, as I am not teaching reasoning.

  26. Briggs


    You’re describing what induction is and what it isn’t. But I can see that we’ll need to discuss this more. Soon, soon.

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