Does the study of math sharpen critical thinking skills?
September 25, 2007 12:07 PM Subscribe
If a man’s wit be wandering, let him post a question about mathematics and reasoning to MetaFilter.
The quote that inspires this question is
“If a man’s wit be wandering, let him study the mathematics.” – Francis Bacon
Is Bacon’s recommendation accurate? Does the study of math sharpen the critical thinking and reasoning skills? I ask because I’m kind of . . . intellectually lazy. I don’t think things through as much as I should. I kind of blurt out opinions based on gut feelings and personal prejudices. I want to get into the habit of thinking and am wondering if there are tools available that will sharpen the mind the way that free weights and kettlebells can sharpen the body. And if math might be one of those tools.
Many thanks in advance.
The quote that inspires this question is
“If a man’s wit be wandering, let him study the mathematics.” – Francis Bacon
Is Bacon’s recommendation accurate? Does the study of math sharpen the critical thinking and reasoning skills? I ask because I’m kind of . . . intellectually lazy. I don’t think things through as much as I should. I kind of blurt out opinions based on gut feelings and personal prejudices. I want to get into the habit of thinking and am wondering if there are tools available that will sharpen the mind the way that free weights and kettlebells can sharpen the body. And if math might be one of those tools.
Many thanks in advance.
Yes. Anything that exercises the brain will sharpen your thinking skills, but math problems often have unique solutions that can be verified; they force you to eliminate the wishy-washy stuff that is such a big part of your love life and MetaFilter.
posted by weapons-grade pandemonium at 12:45 PM on September 25, 2007
posted by weapons-grade pandemonium at 12:45 PM on September 25, 2007
Best answer: Studying mathematics may well help you with the specific areas that you list. Specifically:
(1) I ask because I’m kind of . . . intellectually lazy.
It is difficult to be intellectually lazy with regard to mathematics, because - unlike disciplines in which interpretation plays a much greater role - there is almost always a correct answer (or answers) and an incorrect answer (or answers) to any given question.
(2) I don’t think things through as much as I should.
Carre to hazard a guess at the square root of 22667121? Guessing, even educated guessing, can only take you so far with respect to mathematics. In most cases, you have to think things through (or use a calculator) to arrive at the correct answer.
(3) I kind of blurt out opinions based on gut feelings and personal prejudices.
See (1) on this. "It's my opinion that 2+2=5" won't get you too far in a discussion about mathematics; whereas "its my opinion that the North fought the US Civil War to secure economic dominance rather than free the slaves" can get you quite far in a discussion about US history. (This doesn't make mathematics any better than history as a discipline, but it does make it more difficult to substitute catchy opinion for substantiated, considered thought).
(4) One additional thing: studying mathematics - and especially statistics - really sharpens one's ability to accept counter-intuitive answers. Intellectual laziness often arises in people who are actually quite intelligent and quick-witted, because they are able to take shortcuts to arrive at plausible interpretations or answers to questions. Often, mathematics presents us with situations in which the 'obvious' solution isn't correct.
posted by googly at 12:53 PM on September 25, 2007 [2 favorites]
(1) I ask because I’m kind of . . . intellectually lazy.
It is difficult to be intellectually lazy with regard to mathematics, because - unlike disciplines in which interpretation plays a much greater role - there is almost always a correct answer (or answers) and an incorrect answer (or answers) to any given question.
(2) I don’t think things through as much as I should.
Carre to hazard a guess at the square root of 22667121? Guessing, even educated guessing, can only take you so far with respect to mathematics. In most cases, you have to think things through (or use a calculator) to arrive at the correct answer.
(3) I kind of blurt out opinions based on gut feelings and personal prejudices.
See (1) on this. "It's my opinion that 2+2=5" won't get you too far in a discussion about mathematics; whereas "its my opinion that the North fought the US Civil War to secure economic dominance rather than free the slaves" can get you quite far in a discussion about US history. (This doesn't make mathematics any better than history as a discipline, but it does make it more difficult to substitute catchy opinion for substantiated, considered thought).
(4) One additional thing: studying mathematics - and especially statistics - really sharpens one's ability to accept counter-intuitive answers. Intellectual laziness often arises in people who are actually quite intelligent and quick-witted, because they are able to take shortcuts to arrive at plausible interpretations or answers to questions. Often, mathematics presents us with situations in which the 'obvious' solution isn't correct.
posted by googly at 12:53 PM on September 25, 2007 [2 favorites]
If you study math rigorously (reading and understanding the proofs, or better yet learning to engage in the art/science of proof yourself) it almost certainly will curb some of the tendencies to not think things through. Flashes of insight, intuition, speculation, and educated guesses are all acceptable methods for exploring territory in the discipline, but at the end of the day, you have to construct a very thorough and rigorous argument in order to make statements people will take seriously. Work enough in constraints like that (mathematical or otherwise) and you'll develop something of a habit.
It might also make you a clearer communicator (you have to learn to write about some pretty abstract concepts rather precisely), better at recognizing pattern and structure and systematizing things, and give you an interesting set of ready tools for analyzing some useful well-studied problem domains. It won't help you with everything, but it really is unreasonably effective and pretty cool, and most of my regrets about studying it in school are that I didn't learn my stuff better than I did.
The other regrets are that studying it meant I didn't get to spend as much time outside, traveling, socializing, or practicing music, but you know, tradeoffs and everything.
posted by weston at 1:05 PM on September 25, 2007 [1 favorite]
It might also make you a clearer communicator (you have to learn to write about some pretty abstract concepts rather precisely), better at recognizing pattern and structure and systematizing things, and give you an interesting set of ready tools for analyzing some useful well-studied problem domains. It won't help you with everything, but it really is unreasonably effective and pretty cool, and most of my regrets about studying it in school are that I didn't learn my stuff better than I did.
The other regrets are that studying it meant I didn't get to spend as much time outside, traveling, socializing, or practicing music, but you know, tradeoffs and everything.
posted by weston at 1:05 PM on September 25, 2007 [1 favorite]
Could you be more specific about what stimulated this admirable resolution, what kind of results you desire?
Free weights strengthen your upper body, but only indirectly benefit your running, and vice versa. So, seek the right level of initial challenges that will yield small but genuine successes in areas that you find interesting. That'll take trial and error. Don't be discouraged by initial sparse results. Years ago, an earnest rich fella asked essentially this same question and got the reply, "There is no royal road." But there is a road! Find like-minded companions for encouragement and pacing. MathForum sponsors several different levels and topic areas of "problem of the week". MathCounts caters to young competitors. Mathpuzzle.com is one of many mathematical recreations pages. Or perhaps you aspire to unsolved problems. Try some popular books on mathematics. As a boy, I got inspiration from Asimov's collections such as Science, Numbers, and I. A library will have many selections.
posted by gregoreo at 1:06 PM on September 25, 2007 [1 favorite]
Free weights strengthen your upper body, but only indirectly benefit your running, and vice versa. So, seek the right level of initial challenges that will yield small but genuine successes in areas that you find interesting. That'll take trial and error. Don't be discouraged by initial sparse results. Years ago, an earnest rich fella asked essentially this same question and got the reply, "There is no royal road." But there is a road! Find like-minded companions for encouragement and pacing. MathForum sponsors several different levels and topic areas of "problem of the week". MathCounts caters to young competitors. Mathpuzzle.com is one of many mathematical recreations pages. Or perhaps you aspire to unsolved problems. Try some popular books on mathematics. As a boy, I got inspiration from Asimov's collections such as Science, Numbers, and I. A library will have many selections.
posted by gregoreo at 1:06 PM on September 25, 2007 [1 favorite]
Any part of mathematics will teach you the formal, rigorous way that mathematicians define concepts and construct a logical argument. But if you want to improve your critical reasoning, study basic probability and statistics.
Probability will teach you how to make a valid assertion about something uncertain. Statistics will teach you how to make a valid generalization when you only know part of the data. Both of them will teach you how to identify the limits of what you can say based on what you know.
Armed with that basic knowledge, you'll be able to spot the exaggerations and flawed reasoning in newspaper articles about science, economics, or society. You'll understand what it really means when your doctor tells you something is not a risk. And you'll be able to recognize debates where both sides are making claims that can't be verified.
posted by fuzz at 1:57 PM on September 25, 2007
Probability will teach you how to make a valid assertion about something uncertain. Statistics will teach you how to make a valid generalization when you only know part of the data. Both of them will teach you how to identify the limits of what you can say based on what you know.
Armed with that basic knowledge, you'll be able to spot the exaggerations and flawed reasoning in newspaper articles about science, economics, or society. You'll understand what it really means when your doctor tells you something is not a risk. And you'll be able to recognize debates where both sides are making claims that can't be verified.
posted by fuzz at 1:57 PM on September 25, 2007
A lot of people are dancing around the point: what mathematics teaches you is rigor. That's what Bacon was talking about.
posted by Steven C. Den Beste at 3:37 PM on September 25, 2007
posted by Steven C. Den Beste at 3:37 PM on September 25, 2007
Absolutely studying math will strengthen your logical skills. You don't have to train for triathlons in order to jog, though - I doubt a peer-reviewed dissertation on invariant transformations in n-dimensional Hilbert space is your ultimate goal here.
Find a good, peer-recommended introductory textbook in calculus (or college algebra, or introductory analysis, or proof-based geometry; depending on what feels interesting to you and where you left off in school). Read it at your own pace and do the problems. It's all too easy to read any text, get a warm fuzzy feeling of comprehension, and skate on. A good math text won't let you do this, because every chapter builds directly on previous arguments - a lazy reader will get very lost very fast. The problems in a math textbook are there exactly to make you slow down and really, really understand every bit of the argument from every angle before you move on. And then the next chapter will take some facet of what you just learned and expand it in a different direction; subsequent chapters may revisit previous chapters in a different light - a good math textbook basically walks you step by step through constructing a huge, varied, logically consistent edifice out of a few basic premises, and makes sure you're able to defend any bit of it in detail. It's useful.
For your day-to-day reason-boosting needs, sudoku, good-quality crosswords (especially British-style), and those "Janey lives next to Tommy who is not the conductor" logic puzzles can be helpful. They're not a full workout, but they're stretching exercises.
posted by ormondsacker at 3:40 PM on September 25, 2007
Find a good, peer-recommended introductory textbook in calculus (or college algebra, or introductory analysis, or proof-based geometry; depending on what feels interesting to you and where you left off in school). Read it at your own pace and do the problems. It's all too easy to read any text, get a warm fuzzy feeling of comprehension, and skate on. A good math text won't let you do this, because every chapter builds directly on previous arguments - a lazy reader will get very lost very fast. The problems in a math textbook are there exactly to make you slow down and really, really understand every bit of the argument from every angle before you move on. And then the next chapter will take some facet of what you just learned and expand it in a different direction; subsequent chapters may revisit previous chapters in a different light - a good math textbook basically walks you step by step through constructing a huge, varied, logically consistent edifice out of a few basic premises, and makes sure you're able to defend any bit of it in detail. It's useful.
For your day-to-day reason-boosting needs, sudoku, good-quality crosswords (especially British-style), and those "Janey lives next to Tommy who is not the conductor" logic puzzles can be helpful. They're not a full workout, but they're stretching exercises.
posted by ormondsacker at 3:40 PM on September 25, 2007
Best answer: From my own field, a doctor named William Osler on this topic:
Math is just as good a field to do work in as any other.
posted by ikkyu2 at 3:46 PM on September 25, 2007 [7 favorites]
I propose to tell you the secret of life as I have seen the game played, and as I have tried to play it myself.… Though a little one, the master-word looms large in meaning. It is the open sesame to every portal, the great equalizer in the world, the true philosopher's stone, which transmutes all the base metal of humanity into gold. The stupid man among you it will make bright, the bright man brilliant, and the brilliant student steady. With the magic word in your heart all things are possible, and without it all study is vanity and vexation. The miracles of life are with it; the blind see by touch, the deaf hear with eyes, the dumb speak with fingers. To the youth it brings hope, to the middle-aged confidence, to the aged repose. True balm of hurt minds, in its presence the heart of the sorrowful is lightened and consoled. It is directly responsible for all advances in medicine during the past twenty-five centuries.… And the master-word is Work, a little one, as I have said, but fraught with momentous sequences if you can but write it on the tablets of your hearts, and bind it upon your foreheads.
Math is just as good a field to do work in as any other.
posted by ikkyu2 at 3:46 PM on September 25, 2007 [7 favorites]
Carre to hazard a guess at the square root of 22667121? Guessing, even educated guessing, can only take you so far with respect to mathematics.
Hate to point this out, but guessing is how you get square roots. Unless the number is a perfect square, it's going to be irrational, and all you can do is estimate it or leave it under the radical sign.
Now, you have picked a perfect square, and I could have figured it out in time rather than narrowing to "about 4500" in two iterations and leaving it at that, but that's not the general case.
posted by darksasami at 3:49 PM on September 25, 2007
Hate to point this out, but guessing is how you get square roots. Unless the number is a perfect square, it's going to be irrational, and all you can do is estimate it or leave it under the radical sign.
Now, you have picked a perfect square, and I could have figured it out in time rather than narrowing to "about 4500" in two iterations and leaving it at that, but that's not the general case.
posted by darksasami at 3:49 PM on September 25, 2007
Math is good for this, so is logic. If no specific branch of math appeals, you could get an elementary logic book (some recommendations here) and work through the problems in it. A formal logic text will be mostly symbols and proofs, not so much real-world reasoning. You could also get a critical thinking textbook (some recs here); these tend to be not super rigorous but full of interesting real-world examples. Depends what motivates you.
Really, any sustained mental engagement with complex material will work -- for example you could try reading through Russell's Problems of Philosophy, or (longer and looser) Gaarder's Sophie's World (a sort of novel-style history of philosophy), and really trying to understand the progression of questions/issues in each. But as said above, logic or math have problem-sets with answer keys in the back, so in that way they may be easier projects to do alone, since you can keep yourself honest.
posted by LobsterMitten at 5:20 PM on September 25, 2007 [1 favorite]
Really, any sustained mental engagement with complex material will work -- for example you could try reading through Russell's Problems of Philosophy, or (longer and looser) Gaarder's Sophie's World (a sort of novel-style history of philosophy), and really trying to understand the progression of questions/issues in each. But as said above, logic or math have problem-sets with answer keys in the back, so in that way they may be easier projects to do alone, since you can keep yourself honest.
posted by LobsterMitten at 5:20 PM on September 25, 2007 [1 favorite]
There are also nice pop science/philosophy/logic essayists you could start with (eg Martin Gardner, Raymond Smullyan, etc). Borges; Godel, Escher Bach, etc are standard recommendations in this vein.
posted by LobsterMitten at 5:25 PM on September 25, 2007 [1 favorite]
posted by LobsterMitten at 5:25 PM on September 25, 2007 [1 favorite]
Bacon died in 1626, years before Descartes had published anything (no analytic geometry), and more than 50 years before Newton's Principia.
I'm not sure how rigorous mathematics was then or was considered to be. When I went looking for a source for the famous line usually attributed to Euler: 'persist my son, and faith will come to you' which I remembered as having been addressed to one of the Bernoulli brothers who was having very serious doubts about the foundations of his chosen field, I ran across this interesting paragraph from a review of a book about the history of calculus:
Calculus in 1800 was in a curious state. There was no doubt that it was correct. Mathematicians of sufficient skill and insight had been successful with it for a century. Yet no one could explain clearly why it worked. To be sure, experts would probably have agreed that some notion of "limit" lay behind derivatives, and of course integrals were defined as antiderivatives and thus raised no separate questions. But the discussions of the foundational issues had been desultory and inconclusive. Students by and large were not instructed in calculus, they were initiated into it. If they were gifted with the right insight, practice would then give them an intuitive feeling for the right results. The motto of the period, attributed (perhaps wrongly) to D'Alembert, was Allez en avant, et lafoi vous viendra:Go forward, and faith will come to you...
Today, while we are clearly in the midst of a great Golden Age of mathematics, I would say the minds of mathematicians are wandering all over the damned place. Consider that mathematics advanced enough to contain ordinary arithmetic cannot be proved to be consistent, and will always contain propositions which cannot be proved true or false, due to Godel, and that a sphere can be decomposed into a finite number of pieces and reassembled into two spheres equal in size to the original with no hollow spots-- Tarski's paradox.
posted by jamjam at 5:44 PM on September 25, 2007
I'm not sure how rigorous mathematics was then or was considered to be. When I went looking for a source for the famous line usually attributed to Euler: 'persist my son, and faith will come to you' which I remembered as having been addressed to one of the Bernoulli brothers who was having very serious doubts about the foundations of his chosen field, I ran across this interesting paragraph from a review of a book about the history of calculus:
Calculus in 1800 was in a curious state. There was no doubt that it was correct. Mathematicians of sufficient skill and insight had been successful with it for a century. Yet no one could explain clearly why it worked. To be sure, experts would probably have agreed that some notion of "limit" lay behind derivatives, and of course integrals were defined as antiderivatives and thus raised no separate questions. But the discussions of the foundational issues had been desultory and inconclusive. Students by and large were not instructed in calculus, they were initiated into it. If they were gifted with the right insight, practice would then give them an intuitive feeling for the right results. The motto of the period, attributed (perhaps wrongly) to D'Alembert, was Allez en avant, et lafoi vous viendra:Go forward, and faith will come to you...
Today, while we are clearly in the midst of a great Golden Age of mathematics, I would say the minds of mathematicians are wandering all over the damned place. Consider that mathematics advanced enough to contain ordinary arithmetic cannot be proved to be consistent, and will always contain propositions which cannot be proved true or false, due to Godel, and that a sphere can be decomposed into a finite number of pieces and reassembled into two spheres equal in size to the original with no hollow spots-- Tarski's paradox.
posted by jamjam at 5:44 PM on September 25, 2007
Jamjam, Euclid essentially wrote the book on "rigor". Euclidean Geometry was the ideal at that time to which all other mathematics and all science and philosophy tried to match.
Yes, Bacon knew all about rigor.
posted by Steven C. Den Beste at 9:26 PM on September 25, 2007
Yes, Bacon knew all about rigor.
posted by Steven C. Den Beste at 9:26 PM on September 25, 2007
That's an excellent point about Euclid, SCDB. The book review I quote above goes on to say:
Besides, mathematicians still remembered the tradition of proof that was their proud inheritance from the Greeks. To establish something "in the style of geometry" was a byword for establishing it beyond doubt. Particularly galling was the fact that Archimedes had established some "calculus" results in exactly that style.
Euclids Elements continued to be used on English schoolboys (I was tempted to say 'against') into the 20th century, a practice which Bertrand Russell, among others, found disgraceful. He has this to say about Euclid as a model of reasoning:
It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid's earlier proofs fail before this test.
As far as Bacon is concerned, I take your word for it that he "knew all about rigor," but I would deny that he was "all about rigor." My favorite among his aphorisms is "there is no excellent beauty that hath not some strangeness in the proportion."
The thrust of my answer is to resist the conception of mathematics as a kind of sanitorium of the intellect, to which minds exhausted and unhealthy may repair for rehabilitation; I think of it instead as one of the chief habitats of those aspects of the universe which are 'stranger than we can suppose,' into which we may venture only at some peril to our wishful conviction that we live in a safe and orderly world.
And along these lines, why pursue rigor? You will achieve it soon enough whether you want to or not.
posted by jamjam at 12:07 PM on September 26, 2007
Besides, mathematicians still remembered the tradition of proof that was their proud inheritance from the Greeks. To establish something "in the style of geometry" was a byword for establishing it beyond doubt. Particularly galling was the fact that Archimedes had established some "calculus" results in exactly that style.
Euclids Elements continued to be used on English schoolboys (I was tempted to say 'against') into the 20th century, a practice which Bertrand Russell, among others, found disgraceful. He has this to say about Euclid as a model of reasoning:
It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid's earlier proofs fail before this test.
As far as Bacon is concerned, I take your word for it that he "knew all about rigor," but I would deny that he was "all about rigor." My favorite among his aphorisms is "there is no excellent beauty that hath not some strangeness in the proportion."
The thrust of my answer is to resist the conception of mathematics as a kind of sanitorium of the intellect, to which minds exhausted and unhealthy may repair for rehabilitation; I think of it instead as one of the chief habitats of those aspects of the universe which are 'stranger than we can suppose,' into which we may venture only at some peril to our wishful conviction that we live in a safe and orderly world.
And along these lines, why pursue rigor? You will achieve it soon enough whether you want to or not.
posted by jamjam at 12:07 PM on September 26, 2007
jamjam, is your point that intuition as well as rigor has a role in math (I agree), or is it that rigor has no role in math (I disagree very heartily)?
posted by LobsterMitten at 1:45 PM on September 26, 2007
posted by LobsterMitten at 1:45 PM on September 26, 2007
Response by poster: Could you be more specific about what stimulated this admirable resolution, what kind of results you desire?
(Why, thank you!) It was MetaFilter, actually. I joined after a long period of intellectual inactivity, of not reading books, of sleepwalking through jobs that made no mental demands on me. I joined, I jumped into the discussions and found myself outclassed on a few occasions. No surprise there. They just had better conditioning, sharper thinking skills than I did.
Around the same time, I started working out pretty seriously. I liked the results I got from hard work. My body felt fluid and graceful. I performed daily tasks much more easily. It occurred to me that maybe my mind could benefit from a similar dose of stimulation and discipline.
I remembered the quote above and, not knowing any mathematicians in real life, thought I'd ask the MeFites what they thought about it.
And I've been extremely happy with the answers I've gotten here, as always. Thank you all very much for taking the time to share your expertise and experience! Again, thank you!
posted by jason's_planet at 6:35 PM on September 26, 2007
(Why, thank you!) It was MetaFilter, actually. I joined after a long period of intellectual inactivity, of not reading books, of sleepwalking through jobs that made no mental demands on me. I joined, I jumped into the discussions and found myself outclassed on a few occasions. No surprise there. They just had better conditioning, sharper thinking skills than I did.
Around the same time, I started working out pretty seriously. I liked the results I got from hard work. My body felt fluid and graceful. I performed daily tasks much more easily. It occurred to me that maybe my mind could benefit from a similar dose of stimulation and discipline.
I remembered the quote above and, not knowing any mathematicians in real life, thought I'd ask the MeFites what they thought about it.
And I've been extremely happy with the answers I've gotten here, as always. Thank you all very much for taking the time to share your expertise and experience! Again, thank you!
posted by jason's_planet at 6:35 PM on September 26, 2007
And along these lines, why pursue rigor?
Because that is what the OP is seeking, though he didn't know it.
posted by Steven C. Den Beste at 8:52 PM on September 26, 2007
Because that is what the OP is seeking, though he didn't know it.
posted by Steven C. Den Beste at 8:52 PM on September 26, 2007
Response by poster: Because that is what the OP is seeking, though he didn't know it.
Indeed.
posted by jason's_planet at 9:30 PM on September 26, 2007
Indeed.
posted by jason's_planet at 9:30 PM on September 26, 2007
For a discussion along these lines, check this FPP.
posted by daksya at 1:14 AM on September 27, 2007
posted by daksya at 1:14 AM on September 27, 2007
This thread is closed to new comments.
posted by pammo at 12:44 PM on September 25, 2007