Slashdot Mirror
Mathematics As the Most Misunderstood Subject
Lilith's Heart-shape writes "Dr. Robert H. Lewis, professor of mathematics at Fordham University of New York, offers in this essay a defense of mathematics as a liberal arts discipline, and not merely part of a STEM (science, technology, engineering, mathematics) curriculum. In the process, he discusses what's wrong with the manner in which mathematics is currently taught in K-12 schooling."
Yes, the problem teaching Math(s) and programming (applied Math(s)) is that it's just about intelligence - which you can't teach. The smarter you are, the better you'll be able to figure it out. The problem with teaching is all the generalists who think because they have a "Degree in Education" they are able to teach any topic. Traditionally dry disciplines need to be taught by specialists with passion and enthusiasm for their topic, not by generalists who happen to have a gap in their timetable.
let A=1, B=1
A^2=B^2 because A=B, so
A^2=AB and
A^2-B^2=A^2-AB , next we factor
(A+B)(A-B)=A(A-B) , divide like terms
(A+B)=A
substituting our variables for their values we learn that
2=1.
The brain can be trained and the processes of problem-solving can be generalised - see Polya's How to Solve It. But it doesn't help much to just read the book: you've got to practice, and practice, and practice some more. You must make mistakes and learn from them. You must be prepared to accept multiple inputs rather than merely those which reinforce your strengths and/or prejudices. You must sometimes, as the old 9/11 troll used to say, get some perspective - don't count the angels on a pinhead while Rome burns, even while the most secure of academic positions involves the former and there's such an alluring spirit of mental masturbation in many disciplines and departments.
Meanwhile a good teacher has spent enough decades on some area that he knows both where to provide you hints on specific complex problems and which direction to guide you in when you're contemplating your whole professional life. But, again, don't just choose the teacher who happens to share your academic and ethical prejudices.
I can attest that "true" math is very removed from computation. The computational classes are all regarded as the "easy" classes. This is in contrast to the "hard" classes, real analysis and abstract algebra. Being thrown into real analysis after just one quarter of study in proofs is extremely rough going. If proofs were introduced as puzzles or just introduced earlier in education the whole of America would be better off for it.
My own motivations for being in math are for the challenge and because of the lack of concrete answers in calculus. Trigonometric functions especially are always treated as little boxes that magically calculate what you need.
In any case, at least math attracts the curious.
Eat sleep die
>>Mathematics is the foundation for philosophy
Eh, kinda. Advanced logic is the foundation for a lot of modern philosophy, but Wittgenstein and the rest of the 20th century analytics were just responding to the tremendous success of physics at figuring shit out, and wanted to smear some of that patina on themselves. Well, logic has always been a part of philosophy (think Socrates and his syllogisms) but reading the Tractatus is like reading a modern computer science proof.
Which isn't surprising, either, given that computer science is essentially applied philosophy in a lot of ways. (cf Bertrand Russell, etc.) If you've ever sat through a class where philosophers have sat there talking themselves in circles about how an object can't both be is-a and has-a at the same time, you (if you're like me) feel like leaping up and just telling them to fucking encode whatever paradox they're trying to create in a object hierarchy, and be done with it. I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers".
It does kind of bug me though, that a person who graduates with a degree in mathematics (which is a fairly difficult, hard-nosed subject) gets a wishy-washy BA degree, whereas a hippie with a degree in "environmental engineering" gets a BS, but ultimately I think there's a lot of problems with our current conception with categorizing things into "science" and "not-science". Economics and Climatology are very analogous in terms of what they do - gathering tons of data, running analyses on it, and projecting things out into the future, and both are essentially "empirical studies of the world about us" (i.e. a sort of base level of science, though with the testing, replication and confirmation bits left out), but we consider one to be a social science and another to be hard science. There's also a huge debate now over Anthropology, after the American Anthropology Association dropped "science" from its official bits.
Basic math is easy enough for nobody to have an excuse for not knowing it.
No sig today...
(A+B)(A-B)=A(A-B) , divide like terms
Divide by zero error! After this point, every conclusion is invalid since the results are undefined.
Depressingly, some people (adults as well as kids) would not spot that.
Those who can make you believe absurdities can make you commit atrocities. - Voltaire
Okay.
Philosophy is the process of speaking greek and stroking beards. Therefore by stroking a grecians beard, I shall become a philosopher.
I have defined philosophy (badly) and applied a complete absense of logic. Is that not what you meant?
I've seen the following link in many a Slashdot thread before, but it certainly bears repeating here: "A Mathematician's Lament" by Paul Lockhart It's mostly known as an insightful critique of what's wrong with K-12 math education, but I've always liked it as an explanation of why people who enjoy math do it in the first place: it's satisfying in an artistic way. I think it would be great if more students saw math as something worth doing for its own sake, like art or athletics, and hey, it lets you do science and engineering too.
In fact, this summary sounds similar enough to "Lament" that I wouldn't be surprised if this Dr. Lewis was inspired by and/or cited it. But this is Slashdot, so I'll let someone else check that out.
"This algorithm runs in constant time. Come on, 2,147,483,648 is a constant..."
This is exactly the kind of thinking that has got us into the mess we're into now.
Learning math is just as difficult as learning any other subject or content material. Deciphering poetry, learning programming, studying psychological theory, and learning calculus all involve concentration, study, and struggle from the learner. No one is born knowing any of those things, therefore they all must be learned. The entire point of the OP is to say that the way we go about teaching math is wrong and that people need to reconceptualize how they teach the information because it doesn't make sense to the learner. In the end, its all difficult to some degree. It's when you have that "A-Ha!" moment, it clicks, and you get it. But if you have some terrible algebra teacher who doesn't understand advanced math or someone who doesn't care that you learn, only that you can complete problems 1-50 in a mechanic fashion, then of course it's going to seem difficult (or more difficult than it should be).
Carl Sagan quotes get you an automatic +5 on all posts.
I have a cousin who is great at mathematics, and really can see mathematics as an art. Whereas I am happy if I can solve a problem, he will look for an "elegant solution". I had a number of equations that I solved, trying to optimise the buffer size for various input queues. I shown him, and he quickly said that I had the right answer. A day later he came and shown me how he derived an equation that could simply solve all problems of this type. He also generalised it to allow buffer sizes that were complex numbers. The first part was very useful to me, the second absolutely useless - but to him it was all just interesting.
This is one way that mathematics as an art is unlike any other art. It gives useful results. I have heard time and time again about engineers going to the mathematics department of a University asking how they can solve a "new" problem - to be told that the solution had been discovered a century before. I am sure most of these solutions came from someone just wanting to find an elegant way of expressing something without thought of any use. So if its an art and is useful why do so few people follow it?
The answer is obvious, because its hard! In many forms of art you can slap anything down and convince someone that it has value and its art. This may not always have been true, before photography accurate representational art was highly valued - but today someone producing a lifelike portrait will not be values as much as someone slapping their name on an unmade bed! Mathematics has to be right, you can't just slap down a few numbers and call it an equation. This is the basic problem that anyone will have in persuading someone to follow maths for its art, there are a lot easier ways to become an artist.
Well, we would likely all be malnourished, due to lack of fertilizers, at least those of us who hadn't died at childbirth or soon after. There wouldn't be an Internet to talk on, but that would be okay, since we wouldn't have time to use one due to the lack of engines and the resulting need to do backbreaking labour 16 hours a day. In short, our lives would be miserable, but due to lack of medicine, they would at least be short.
Missing these kinds of little details is why I have very little respect of philosophers. As far as I can tell, most of them chose their field because it doesn't punish sloppy work. And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.
Philosophy means you accept the human condition. Technorcacy means you try to do something about it. Hope for a better world in the future lies on the latter, not the former.
Forget magic. Any technology distinguishable from divine power is insufficiently advanced.
I wish science in general was considered part of what a learned person has to know. I mean, if you want to pass for an intellectual you have to read your Dante, your Beckett and you at least need to know who Lautreamont was. But, apparently, you can very well get away with thinking that you can suck gravity out of a room the way you suck air, or with not having even heard about string theory. That divorce makes no sense, and it was impossible in the history of ideas till very recently. And Euler's formula is more beautiful than most poems.
The way math is taught in schools is atrocious. Most math texts that I've used with 5th and 6th graders emphasize learning processes and methods for solving a set of problems. The texts do not hold all of the blame, however. The texts are written to follow state and national standards. The standards are written in such a way to emphasize process and not necessarily apprehension of greater concepts. For example:
5th Grade Level Expectation 1. Differentiate between the term factor and multiple, and prime and composite (N-1-M)
While these vocabulary items are important and these skills are definitely useful, learning this skill in isolation (which most texts teach) is pretty useless as students do not connect these skills to a greater picture.
A revision of mathematics standards and teaching methods will go a long way to improving the quality of mathematics education. A holistic approach that includes some wrote learning of basic skills and lots of real application problems. Real application problems are not word problems. How many "real" word problems have you had to solve in the last ten years?
Some texts such as Every Day Math from the University of Chicago does a much better job at integrating all sorts of skills and teaching in a much more holistic method. It includes some excellent modeling exercises, games that rely on a real understanding of mathematical principals for mastery and interesting lessons. But even the best text can't help a kid if they don't have a good teacher that really understands mathematics. Watching an uniformed teacher try to explain what a prime number is, or a different method for division (such as repeated subtraction) is painful. They simply can't do it. Unfortunately, in my experience most of the teaching candidates that were in my classes thought that math was "hard" and "didn't really matter." They scraped by with the lowest possible scores in the required math classes and one even told me she "wasn't going to bother teaching math." While this is pure anecdotal evidence, the declining math scores in the US show that we really do suck and producing math teachers.
The problem stems from bad math teachers badly teaching math which of course leads to more poorly instructed math teachers. Placing a real emphasis on reading and mathematics, with highly qualified and well-supported specialists is the only way we're going to solve this problem. Unless we have some real political will akin to that found during the space race, we're not going to solve this problem any time soon. We'll just keep cranking out kids that think that math is done by computers and a few nerds that wave their magic math wand over problems to find solutions.
This one's tricky. You have to use imaginary numbers, like eleventeen... --Hobbes
While agriculture requires backbreaking labour, hunter-gatherer societies only worked a couple of days a week. Not that I advocate a return to it, but backbreaking labour all the livelong day was not universal in ancient society.
Philosophical journals have the same rigorous standards for papers as journals for the various sciences. Your view of philosophy is about as valid as a grizzled mountain man who mutters about hard science being all book-learnin' and mumbo-jumbo.
Even that is a statement of philosophy. Furthermore, you seem unaware that many calls for improving human lives came from works of philosophy: More's Utopia, Kirkegaard's questions of metaethics, even what is often called the beginning of the Western tradition, when Socrates hung out in the agora and asked passersby "What if what you comfortably believe is wrong?"
Economics and Climatology are very analogous in terms of what they do - gathering tons of data, running analyses on it, and projecting things out into the future, and both are essentially "empirical studies of the world about us" (i.e. a sort of base level of science, though with the testing, replication and confirmation bits left out), but we consider one to be a social science and another to be hard science.
Well, economics is, especially in its present state, largely influenced by individuals, who can be a lot harder to predict than wind currents. You may identify trends, constants and correlations, but mostly in hindsight. Accurate predictions are as scarce as in cartomancy and useful controlled experiments are hard to imagine. While Climatology shares some of those characteristics, I think we have a much higher chance of predicting a storm than the stock market. Unless tons of people start walking around with nuclear powered, oversized fans. If you catch my drift.
Philosophy means you accept the human condition.
No.. Philosophy means questioning the human condition. it's confronting the status quo and asking "why?"
So exactly the opposite in every way of what you think it is.
You're also wrong in your assumption that philosophy and technocracy are mutually exclusive, in fact if they aren't mutually inclusive, then as a technocrat you're trying to find solutions when you don't even know what the problem is.
Philosophy is a very powerful way of thinking, and in no way whatsoever does it represent conformity or acceptance, it represents freedom of thought to think critically.
In fact philosophy really should be tought in schools, it's the basis of how we view the world today, and if the future is bright, it will be philosophy to thank and the people who dared to question the status quo.
Why are people even debating philosophy vs technocracy? Why should someone have to choose one over the other? How do people get dragged into such nonsense? Here a new subject for you: tomatoes vs rainbows. Go.
If you've ever sat through a class where philosophers have sat there talking themselves in circles about how an object can't both be is-a and has-a at the same time, you (if you're like me) feel like leaping up and just telling them to fucking encode whatever paradox they're trying to create in a object hierarchy, and be done with it. I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers".
I understand where you're coming from, but for many philosophers, what they're doing is not just trying create a practical solution to a problem, but describe reality. Your object model might solve the problems from your point of view, but it includes many built in assumptions about the thing modeled.
In a related way Wittgenstein later came to criticize the Tractatus. Part of the criticism is that if you assume the universe can be fully described with formal logic (logical atomism), then you are already subscribed to a certain type of metaphysics.
Missing these kinds of little details is why I have very little respect of philosophers. As far as I can tell, most of them chose their field because it doesn't punish sloppy work. And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.
There's plenty of philosophy-types who think that Searle is an idiot, too, for the Chinese Room and other things. Guy loves to position himself as a defender of rationality and realism because it lets him belittle poststructuralists with oversimplifications and straw men while acting like a hero of a scientific worldview that he clearly doesn't know that much about.
In some ways his antagonistic materialsm is quite similar to your dismissal of philosophy in general, actually.
You should have had Mr Burton, my maths O level teacher. He was brilliant. He was totally passionate about his subject and he was also a fantastic teacher. he encouraged us to think about maths rather than to just blindly follow formulae. I still vividly remember the lesson where he taught us differential calculus from first principles.
He encouraged us to study outside of lesson time and his door was always open during lunch, or after school. almost every one in his class passed their maths O level with at least a B, over half had A's
It's no exageration to say I owe my career as a developer to him and his enthusiastic teaching.
And the people shall be oppressed, every one by another, and every one by his neighbour Isaiah 3:5
"Meanwhile we have turned the majority of Western humans from independent men into chair-warming consumers singing in lockstep for trinkets."
I suggest you take off your rose coloured glasses and go read some history, in particular just how "free" your average serf was in feudal times and even later. Don't like what your overload or king does? Tough. Complain and you'll probably at best end up homeless or at worst end up swinging from a tree.
People in the west have NEVER been as free as they are now.
So get yourself a fucking clue!
Missing these kinds of little details is why I have very little respect of philosophers.
They don't "miss" those details, they're not in scope.
As far as I can tell, most of them chose their field because it doesn't punish sloppy work.
Philosophy does punish sloppy work. relentlessly. Philosophical work is subject to more scrutiny and criticism than any discipline I know of, and that includes pure maths.
And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.
Laughing something out doesn't work in philosophy. Unlike whatever discipline you work in, it seems, in philosophy you have to show the reasons why something is wrong. And if you think the issue of emergent properties hasn't been considered in excruciating detail in connection with Searle's Chinese Room thought experiment then you clearly have no idea what philosophy is doing.
Philosophy means you accept the human condition.
Say what? Some philosophy is abstract, but so is some maths. Lots of philosophy (philosophy of science, political philosophy, ethics) is concerned with changing the human condition. Maybe you criticise philosophy because it didn't discover antibiotics (although it did lay a lot of the foundations), but do you criticise biology because it didn't invent democracy? Both changed the human condition, in ways appropriate to their respective disciplines.
Quidnam Latine loqui modo coepi?
I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers"
Well, go on then, if it's that fucking simple and obvious. Put those silly old philosophers in their place, what do they know?
I'm thinking of writing a book called "Why do so many students of Computer Science think they have solved all the riddles of the universe because they know how to write a sorting algorithm?"
To have a right to do a thing is not at all the same as to be right in doing it
A Ph.D. tells you nothing except that the holder did some original research at an early point in their career.
There is also little if any correlation in being able to research, and being able to teach. Culturally, "everyone knows" the purpose of a phd is to become a professor and teach university students while collecting a $100K+ salary. The upper 50% to 10% cream of the crop actually get hired to do that. So, pretty much by definition, as a general cross section of the population, they are in the bottom of the barrel of teaching ability. So I'd be expecting, unless they're education phds, they're almost by definition probably not going to be good teachers.
"Science flies us to the moon. Religion flies us into buildings." - Victor Stenger
Philosophy is only a pretty word for wild speculation/daydreaming/brainstorming
You are exactly right, if your definition of "philosophy" is "wild speculation/daydreaming/brainstorming".
Unfortunately for you, words have generally agreed upon meanings, not just whatever brainshite you happen to vomit forth.
To have a right to do a thing is not at all the same as to be right in doing it
In mathematics it is the truthiness of the statement creates "credit" and then we search back in history to find who said it first and then we give the credit to him/her and that is how reputation/respect is created. It flows back in time. Credibility accrues from the statement to the speaker.
In philosophy a bunch of people agree that some one was/is a great philosopher and so they give more value to a statement from such person. The credibility flows from the speaker to the statement.
sed -e 's/Chuck Norris/Rajnikant/g' joke > fact
Here, let me show you an even more beautiful mathematical paradox:
We try to solve this equation: x^2 - x + 1 = 0
We do that by adding x - 1 on both sides: x^2 = x - 1
We multiply both sides by x: x^3 = x^2 - x
Add 1 on both sides: x^3 + 1 = x^2 - x + 1
Recognize the first equation in the right side: x^3 + 1 = 0
Subtract 1 on both sides: x^3 = -1
Take the cube root on both sides: x = -1
Check the answer: (-1)^2 - -1 + 1 = 0
Have fun!
Well, part of it does and another part doesn't.
Confucius say, "Find worm in apple - bad. Find half a worm - worse."
That happens a lot in mathematics too - it has to, or mathematicians would have to spend all their time refuting amateur "proofs" of famous open problems.
Really. Must we contextualize mathematics, or try to talk about what it is or is not? Do we really need to point to a particular cognitive framework as "the reason" why math is not taught "properly?"
To use a slightly loathsome phrase, math "is what it is." Instead of talking about how people should relate to it, I suggest a radical approach: just LEARN it. Teach it for what it is.
I struggled with arithmetic when I was in grade school, not because I didn't understand the rules, but because I kept making mistakes. And my teachers had the wisdom to know that those errors had to be drilled out of me before I could proceed any further. I suffered. I *hated* the tedium. We were asked to multiply two twelve-digit numbers with no assistance from any computing devices or tables; divide four-digit numbers into twenty-digit numbers, until we could do it with 100% accuracy every time. It didn't have to be lightning fast. It just had to be CORRECT.
And when I mastered that skill, it felt fantastic. We moved on to more advanced topics, and each time the teacher made sure we had firmly laid down the next conceptual brick of this vast mathematical edifice we were building for ourselves. It was hard but rewarding. To those critics who might say such an approach would discourage some students, and that some kids just need to be excited by what they learn, clearly you have never really understood what it means to build that foundation. It's got to be ROCK SOLID. No crap about trying to make math "fun" or "interesting" or "relevant." That sort of stuff comes when it comes; they are merely ornaments on the pillars. There's no point in making the structure pretty before you make it sturdy.
So then, how do you get students motivated? It's really quite simple. You challenge them and you force them to bust their asses, and when all their hard work pays off, that sense of accomplishment is better than any drug. To know that you did it on your own, and you have complete confidence in your mastery of the concept, is precisely what must drive them forward. You can't entice them with anything else. You can't try to swaddle the math in some cutesy real-world application, because that is going to be fake, and they know it.
That's the story of how I graduated with my BS in mathematics from one of the most prestigious scientific universities in the world. It was purely the early appreciation for persistence toward understanding mathematics for its own sake. I'm not saying everyone has to keep math "pure." If your goal is to apply it in some other discipline, go for it. But the learning process has to build upon that foundation of math for math's sake.
My older son is in the 2nd grade and is gifted (IQ somewhere around 140). Right now, they're learning simple addition. There's only one problem. He already learned this last year. He was doing complex subtraction with my wife (a teacher) over the summer break. But the class is doing simple addition so that's what he's stuck on.
It gets worse. They're using a so-called "spiral curriculum" this essentially means they learn one way of figuring out that 8+3=11, then learn another way, then a 3rd, 4th and 5th way. My son gets it the first time, yet he has to sit through all of the other ways. He yearns for more advanced math. He asked me about multiplication and division and, when I showed him an example using Legos, he got the concept right away.
He already knows his times tables up to 5 and wants more. But school is boring to him because they don't push him. He isn't being challenged at all. He tends to act out when he's bored too which makes everything more complicated. If you have a child who is falling behind in school, there are resources to help them catch up. If you have a child who is gifted and wants to pull ahead, your kid needs to sit down, be quiet and learn for the fifth time what 8+3 equals.
My sci-fi novel, Ghost Thief, is now available from Amazon.com.
No, they debate fundamental questions (phrased in CS-speak): "Is a pointer to an object the same thing as the object?"
From a CS perspective, the answer is obvious, as is the relationship between a pointer and an object. But philosophers fill up books on this subject.
Just because the lack of a final answer to a problem is dissatisfying does not mean that there must be one. Some problems simply cannot be resolved absolutely.
The example Sartre used is a good one. How would you make a decision in that instance which ends the debate?
The world does not conform to your desire for resolution.
const int one = 65536; (Silvermoon, Texture.cs)
SJW, n: "Someone I don't like, and by the way I'm a fuckwit" - AC
>>I doubt philosophers give a rats ass about pointers, let alone fill up books on the subject.
From the Stanford Encyclopedia of Philosophy:
* Almog, J., J. Perry, and H. Wettstein (eds.) (1989), Themes from Kaplan, New York: Oxford University Press.
* Bach, K. (1987), Thought and Reference, Oxford: Oxford University Press.
* Bach, K. (2004), 'Points of Reference,' in Bezuidenhout & Reimer (eds.) 2004. [Preprint available online]
* Barcan Marcus, R. (1947), "The Identity of Individuals in a Strict Functional Calculus of Second Order," Journal of Symbolic Logic, 12(1): 12-15.
* Barcan Marcus, R. (1961), 'Modalities and Intentional Languages,' Synthese, 13(4): 303-322.
* Barcan Marcus, R. (1993), Modalities, Oxford: Oxford University Press.
* Bezuidenhout, A., and Reimer, M. (eds.) (2004), Descriptions and Beyond, Oxford: Oxford University Press.
* Brandom, R. (1994), Making it Explicit. Cambridge MA: Harvard University Press.
* Brueckner, A. (1986), 'Brains in a Vat,' Journal of Philosophy, 83: 148-167.
* Davidson, D. (1984), Inquiries into Truth and Interpretation, Oxford: Clarendon Press.
* DeRose, K. (2000), 'How can we know that we are not Brains in Vat?,' Southern Journal of Philosophy, 39: 121-148.
* Devitt, M. (1981), Designation, New York: Columbia University Press.
* Devitt, M. (1990), 'Meanings just ain't in the head,' in Meaning and Method: Essays in Honor of Hilary Putnam, Cambridge: Cambridge University Press, pp. 79-104.
* Devitt, M. (1996), Coming to our Senses, Cambridge: Cambridge University Press.
* Devitt, M. and Sterelny, K. (1999), Language and Reality (2nd edition), Cambridge MA: MIT Press.
* Devitt, M. (2004), 'The Case for Referential Descriptions,' in Bezuidenhout and Reimer (eds.) 2004.
* Donnellan , K. (1966), 'Reference and Definite Descriptions,' Philosophical Review, 75: 281-304. [Post-print online version]
* Donnellan, K. (1972), 'Proper Names and Identifying Descriptions,' in D. Davidson and G. Harman (eds) The Semantics of Natural Language, Dordrecht: Reidel.
* Evans, G. (1973), 'The Causal Theory of Names,' Proceedings of the Aristotelian Society, Supplementary Volume 47: 187-208.
* Evans, G. (1982), The Varieties of Reference, Oxford: Oxford University Press.
* Field, H. (2001), Truth and the Absence of Fact, Oxford: Oxford University Press.
* Fodor, J. (1990), A Theory of Content and other Essays, Cambridge MA: MIT Press.
* Frege. G. (1893), 'On Sense and Reference,' in P. Geach and M. Black (eds.) Translations from the Philosophical Writings of Gottlob Frege, Oxford: Blackwell (1952).
* Kaplan, D. (1989), 'Demonstratives: An Essay on the Semantics, Logic, Metaphysics, and Epistemology of Demonstratives and Other Indexicals.' In J. Almog, J. Perry, and H. Wettstein (eds.), Themes from Kaplan, Oxford: Oxford University Press.
* Kripke, S. (1977), 'Speaker's Reference and Semantic Reference,' Midwest Studies in Philosophy 2: 255-76.
* Kripke, S. (1980), Naming and Necessity, Cambridge: Harvard University Press.
* Meinong, A. (1904), 'The Theory of Objects,' in Meinong (ed.) Untersuchungen zur Gegenstandtheorie und Psychologie, Barth: Leipzig.
* Mill, J. S. (1867), A System of Logic, London:
They do, it's called a university and the school of hard knocks. Now we could refine that and go with the European system of dual (or triple) tracks for secondary education but that would be admitting that some snowflakes are less special than others.
There are 4 boxes to use in the defense of liberty: soap, ballot, jury, ammo. Use in that order. Starting now.
Let me try to explain why this appears to work but doesn't. The problem is with this line:
We multiply both sides by x: x^3 = x^2 - x
When solving an equation, there is an assumed logical progression. Suppose you want to solve:
Then, you want to find the set S1={x: x is a solution of (1)}. You do this by transforming the equation repeatedly until you get to a form from which it is easy to derive the solutions. But when you make a transformation of the equation, you need to think about what the set of solutions is after the transformation. Let proposition P1 = x is an element of S1. (Similarly Pn for Sn). If, as the next step, you write:
you are implicitly stating that:
(<=> means "if and only if") If you then write:
x^3 = x^2 - x, (3)
the set of solutions has changed: -1 is introduced as a new solution. In this case, this is because (2) was multiplied by x, which is not a non-zero constant, and thus the meaning of the equation has changed. Logically, you are now stating that:
In other words, if you find an x for which P2 is true, then P3 will also be true for that x, but not the other way round.
Normally when you solve an equation, you implicitly create a progression P1<=>P2<=>...<=>Pn. From this, if you can see that Sn is the set of solutions for (n), then going back by implication from Pn to P1 you can conclude that Sn=S1. However, if the chain is broken and you write P1<=>P2<=>...<=>Pj=>P(j+1)<=>...<=>Pn, you can only conclude that S1 is a subset of Sn. However, because you are missing an implication from P(j+1) to Pj, you cannot say that Sn=S1.
There are many operations that potentially change the set of solutions, such as multiplication of both sides by zero, squaring both sides, and others. At every transformation, you must make sure that the solutions stay the same. In solving other problems, the logical progression can become more complex and then cannot be implicitly assumed like this. Generally, it is always a good idea to know precisely what you are stating in terms of logic.
The mindset of the teacher reminds me of the Harry Chapin song "Flowers Are Red."
Teachers that are that narrow minded should be transferred to places where they can't do any damage to students. Perhaps a prison environment would be best for them. They could at least try to help some of the people they screwed over.