## Mathematics and intellect

For the discussion of math. Duh.

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gorcee
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### Re: Mathematics and intellect

I don't even know what the second law of Thermodynamics is.

I know what the laws of thermodynamics are, but hell if I know (or care) what order they go in, or who declared that order canonical. Same thing with Newton's Laws. Trivia and knowledge are two totally different things, really.

I also have no idea how many electrons are in a potassium atom, because I have no idea where it is on the periodic table. I do know how to acquire that information in very short order, however.

This reminds my of Feynman's story about the map of a cat. He took a graduate biology course for fun once, and while presenting, he started out by naming off some of the anatomy of a cat. Someone interrupted him and said, "but we know all of that already." His thoughts were, well, no wonder it takes you all so damn long to do anything. You spend all your time memorizing stuff that takes 15 minutes to look up.

t0rajir0u
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### Re: Mathematics and intellect

The potassium example is a little different; the position of an element in the periodic table strongly affects its properties and it's not a bad idea to at least have some familiarity with the different orders. Potassium, in particular, is the next alkali metal after sodium.

gorcee
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### Re: Mathematics and intellect

t0rajir0u wrote:The potassium example is a little different; the position of an element in the periodic table strongly affects its properties and it's not a bad idea to at least have some familiarity with the different orders. Potassium, in particular, is the next alkali metal after sodium.

Yes, that I knew, and I know it's on the left. But hell if I know what number it is.

I'm not a chemist, and if I were, it would be useful information to know off-hand. But if you're not a chemist, or working in that field, then probably that information isn't critical to have. And if you're reasonably intelligent, and you know what to do with that sort of information once obtained, then it's really not a big deal to have to google "periodic table" and find the information out in < 60 seconds.

We live in a multi-connected world with nearly-ubiquitous access to unprecedented amounts of information, yet we still put tremendous value on knowing facts rather than knowing what to do with facts.

antonfire
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### Re: Mathematics and intellect

t0rajir0u wrote:Nobody finds it natural to read long strings of existential or universal quantifiers (that I know of).
Perhaps not, but I generally find it more natural to read a series of formulae deriving some equation than to read a bunch English of sentences which do the same thing.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

edahl
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### Re: Mathematics and intellect

notzeb wrote:"A Scheme is a collection of points along with a collection T of collections of those points containing the collection of no points and the collection of all points, with the properties that for any collections of points U, V in the collection T, the collection of all points contained in both U and V is in the collection T, and for any collection of collections that are in the collection T, the collection of points in at least one of these collections is itself a collection T, along with an assignment of a set of things and two operations that we call addition and multiplication on that set that satisfy the distributive, associative, and commutative properties, along with having additive inverses, to each collection of points U in T, so that for any pair of collections U, V in T, U containing every point of V, we have an association of the elements of the set assigned to U to the elements of the set assigned to V respecting the operations of addition and multiplication, and so that...."

That was fun, I might quote you on that one day

mike-l
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### Re: Mathematics and intellect

DavCrav wrote:1) What is the second law of thermodynamics?
2) How many electrons in an atom of potassium?
3) How does a refridgerator work, i.e., cool things down?
4) What is a xylem?
5) What does it mean for a company to be highly geared?
6) What is bauxite?
7) What's the difference between mean and median?

The only ones of those that have anything to do with understanding stuff is 1 and 3. Everything else is terminology. For the record, I have a degree in mathematics and I can only answer 3 and 7 without looking stuff up, but I can find the answer to all 8 in about 10 seconds each on google, which is why it doesn't bother me at all that I don't know them.

One thing that this reminds me though, is that I often get complimented for my excellent memory of things like this, which is a bold face lie, I have a TERRIBLE memory, but I remember how things work. So I'll never be able to memorize wikipedia's text of the 2nd law of thermodynamics
Spoiler:
The entropy of an isolated system consisting of two regions of space, isolated from one another, each in thermodynamic equilibrium in itself, but not in equilibrium with each other, will, when the isolation that separates the two regions is broken, so that the two regions become able to exchange matter or energy, tend to increase over time, approaching a maximum value when the jointly communicating system reaches thermodynamic equilibrium.

But, having now read that (and amazingly I think I've not ever seen that before, it turns out I only knew the 1st), and please correct me if I'm wrong, I should easily remember and be able to recreate most of the wording by remembering that it's really just "entropy increases" worded longly by using the fact (also new to me) that the entropy is highest at equilibrium. Those are two very small facts that my terrible memory is capable of holding and I can reconstruct the text (or something very near that) with just them.

The point of all that is that working on math gets you to think like that, remembering the important parts and being able to build off there.
addams wrote:This forum has some very well educated people typing away in loops with Sourmilk. He is a lucky Sourmilk.

mouseposture
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### Re: Mathematics and intellect

gorcee wrote: You spend all your time memorizing stuff that takes 15 minutes to look up.

Leo Szilard, after moving from physics to biology, complained that he could no longer work in the bathtub, because he kept having to climb out in order to look up a fact. My point being this: in fields like biology, you have to import facts from the reference work into your brain, at least temporarily, in order to do any intellectual work. Some of that information lingers, because, well, that's how memory works. So there's a residue of knowlege which, while not actually needed to do biology well, is, nonetheless, evidence that a person is (strictly: was once) capable of doing it well. Which is why trivia quizzes do tell you something useful, in *some* fields.
... but I do feel that the Internet, by lowering the barrier to "looking it up" is changing this. I wonder how far that trend will go.

Mokele
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### Re: Mathematics and intellect

gorcee wrote:This reminds my of Feynman's story about the map of a cat. He took a graduate biology course for fun once, and while presenting, he started out by naming off some of the anatomy of a cat. Someone interrupted him and said, "but we know all of that already." His thoughts were, well, no wonder it takes you all so damn long to do anything. You spend all your time memorizing stuff that takes 15 minutes to look up.

I love how physicists and mathematicians fail to understand the importance of facts.

Simple scenario: you're performing surgery on a cat, and you nick an artery. You have 20 seconds to find the nearest surgically accessible point upstream to clamp off the blood flow. You're scrubbed up and sterile, thus cannot touch anything not in the sterile field (ie. only your tools). You're in a sterile surgical suite. There are no books or computers (because they cannot be sterilized or cannot survive being sterilized), and no time to have someone leave and check.

We memorize stuff because it's useful to have it instantly at hand, especially if you have no books/internet (in the jungle, in surgery, in a super-urgent situation, etc).

Hands up everyone who wants a surgeon who needs to look everything up.
"With malleus aforethought, mammals got an earful of their ancestor's jaw" - J. Burns, Biograffiti

chilled
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### Re: Mathematics and intellect

erm.... biology versus anatomy...

BlackSails
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### Re: Mathematics and intellect

Surgery isnt a kind of science.

skeptical scientist
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### Re: Mathematics and intellect

t0rajir0u wrote:
DavCrav wrote:1) What is the second law of thermodynamics?
2) How many electrons in an atom of potassium?
3) How does a refridgerator work, i.e., cool things down?
4) What is a xylem?
5) What does it mean for a company to be highly geared?
6) What is bauxite?
7) What's the difference between mean and median?
I think this particular test is a little extreme.

I agree. I could only get five of them. Did anyone besides DavCrav get all 8 without help?
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

isomorpher
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### Re: Mathematics and intellect

I only knew 3 of them. I have never even heard of a xylem or bauxite or filtrum. I think more appropriate questions would be like "How does an airplane wing work?" or "How do you know if a star is going to turn into a black hole?". Questions that are not overly specific, yet require at least a moderate understanding of the topic.

Like it or not, the arts and social sciences are much more tied to our (collective) culture and everyday lives than natural/theoretical sciences or mathematics. So the average person not only will know more about those topics, but they will be able to relate to them better. For people like us where science and math are part of our everyday lives, I think we naturally manifest our perceptions of intelligence with a heavy bias.

I'm trying to be objective as possible, but personally I am subject to that bias and do view science and math as more.... "noble" forms of intelligence.

Marbas
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### Re: Mathematics and intellect

skeptical scientist wrote:
t0rajir0u wrote:
DavCrav wrote:1) What is the second law of thermodynamics?
2) How many electrons in an atom of potassium?
3) How does a refridgerator work, i.e., cool things down?
4) What is a xylem?
5) What does it mean for a company to be highly geared?
6) What is bauxite?
7) What's the difference between mean and median?
I think this particular test is a little extreme.

I agree. I could only get five of them. Did anyone besides DavCrav get all 8 without help?

I am not even half way done with my degree, but I did not. I got 2.

Surgery isnt a kind of science.

I would argue that there is a difference between trades/crafts. And purely academic work. And I would also argue that surgery is at least in part a trade/craft.

I love how physicists and mathematicians fail to understand the importance of facts.

I'm going to agree with this too though. A lot of the mathematics and physics majors I've met don't know how useful it is to have certain things memorized, like say, I'll pick an example I'm familiar with, designing a circuit.
Jahoclave wrote:Do you have any idea how much more fun the holocaust is with "Git er Done" as the catch phrase?

t0rajir0u
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### Re: Mathematics and intellect

antonfire wrote:I generally find it more natural to read a series of formulae deriving some equation than to read a bunch English of sentences which do the same thing.

I have read advice on mathematical writing which disagrees with this, and I personally find computations themselves highly unenlightening whichever way they're presented. (For example, I hate the computational proof of the orthogonality relations for finite groups. The conceptual proof is a million times clearer.) Anything you can do to make a computation seem like something other than mere algebra, in my opinion, helps.

BlackSails
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### Re: Mathematics and intellect

Marbas wrote:I would argue that there is a difference between trades/crafts. And purely academic work. And I would also argue that surgery is at least in part a trade/craft.

Its totally a craft. It uses science much like engineering, but its not a branch of science.

Mokele
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### Re: Mathematics and intellect

chilled wrote:erm.... biology versus anatomy...

What do you think anatomy is studying, rocks? Or maybe, oh, *living things*? You know, things with evolutionary and ecological contexts which govern anatomy, and which anatomy can elighten us about?

BlackSails wrote:Surgery isnt a kind of science.

It's called "making a point", using what is the equivalent of engineering for biology. The point being, you *need* these facts, lots of them, to even be able to function in biology, whether the issue is doing surgery on experimental animals (which, FYI, is a huge part of what we do), figuring out what's safe to grab in the middle of a jungle, or just being able to have an intelligent conversation with a colleague.

All science requires memorization; you memorize Maxwell's equations, I memorize the length-tension relationship of muscle, you memorize the periodic table, I memorize the phylogeny of vertebrates, you memorize how to do a Fourier transform, I memorize how to surgically implant electrodes. And it's not just random data - it's concepts that apply over huge ranges of life. Teach a student the Q10 rule and they have the core of understanding all thermal biology. Teach them the anatomy of a cat and they can find the kidneys in a shark, snake, frog and bird. Teach a student evolution and they can understand pretty much all life.

Just because it's not a formula doesn't mean it's not "a concept". And if physicists had to face the level of diversity we deal with, they'd quickly learn the merits of basic facts.
"With malleus aforethought, mammals got an earful of their ancestor's jaw" - J. Burns, Biograffiti

DavCrav
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### Re: Mathematics and intellect

@Everyone: The point of those questions was to prove a few things:

1) Firstly, they weren't aimed at people who have just finished their degree. They are aimed at those people at the top of their profession (in pretty much any profession) who have been around for a long time and claim to have lots of knowledge about everything because they are in the arts.

2) Some of them might be region-specific, like the xylem question, in the sense that every single child in the UK learns this off by heart.

3) The last question was a bit dodgy, sure. The other seven are all in under-16 school, at least in the UK. I was trying to choose extremely basic questions from a variety of fields. Any biologist will know what a xylem is, sure, but what about their knowledge of the other branches of science?

4) OK, if you know the laws of thermodynamics then I don't care that you don't know which order they go in, you pass!

5) You are never supposed to get 8 out of 8 for a general knowledge test, because it's going to depend on whether my facts and your facts coincide. My point is that will people get any?

6) As for methods versus facts: if you don't know the names of things, how can you tell other people what you mean? Google is all well and good, but what if you are, I don't know, not at your computer? (Or your iPhone.) Maybe you want the feeling that you know things, rather than just having the ability to ask someone else. If this isn't the case, what do you actually need to know at all?

antonfire
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### Re: Mathematics and intellect

t0rajir0u wrote:
antonfire wrote:I generally find it more natural to read a series of formulae deriving some equation than to read a bunch English of sentences which do the same thing.

I have read advice on mathematical writing which disagrees with this, and I personally find computations themselves highly unenlightening whichever way they're presented. (For example, I hate the computational proof of the orthogonality relations for finite groups. The conceptual proof is a million times clearer.) Anything you can do to make a computation seem like something other than mere algebra, in my opinion, helps.
Perhaps, but I suspect you still prefer "the probability density at [imath]x[/imath] of a normally distributed random variable centered at [imath]x_0[/imath] with variance [imath]\sigma[/imath] is [imath]\frac{1}{\sqrt{2\pi\sigma}} e^{(x-x_0)^2/\sigma}[/imath]" to "the probability density, at some given position, of a normally distributed variable is one over the square root of two pi times the variance of the distribution, multiplied by the exponential evaluated at one over the variance times the square of the distance between where the distribution is centered and the given position".

In this case (and in many other cases) I think the notation makes it denser and easier.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

mouseposture
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### Re: Mathematics and intellect

Mokele wrote: There are no books or computers (because they cannot be sterilized or cannot survive being sterilized), and no time to have someone leave and check..

Surgeons routinely use computers while scrubbed in. For example, stereotactic targeting software already allows neurosurgeons to "see" structures from a reference anatomical atlas overlayed on the patient they're operating on. Surgical robotics is carrying this much further. Soon -- probably within our lifetime -- your hypothetical surgeon will *see* that upstream clamp site flashing bright green on her headsup display before she even has a chance to remember it.

The reason (I believe) that facts are important is that human minds require facts to think with. You turn them around in your mind, see how they fit together ... and that's how you come up with new ideas. I can't imagine a completely empty mind, equipped with nothing but logic, arriving at any conclusions at all. I suspect it's impossible, but I'll make the weaker claim that it isn't *human*

t0rajir0u
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### Re: Mathematics and intellect

Certainly I wasn't trying to suggest you abandon mathematical notation altogether. But many people are under the impression that if you write something in English, it's not math (especially people who haven't had much experience with proof-based mathematics), and this is a terrible way to think about writing math.

Anyway, I should probably cite my sources here; Munkres' advice is here and Knuth's advice is here. Admittedly it is less extreme than I remember, but look in particular at the second section of Knuth's.

Mokele
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### Re: Mathematics and intellect

mouseposture wrote:Surgeons routinely use computers while scrubbed in. For example, stereotactic targeting software already allows neurosurgeons to "see" structures from a reference anatomical atlas overlayed on the patient they're operating on.

Yes, but for those of us without $40 million dollar surgical suites, it's not exactly a good idea to put a laptop through the autoclave. "With malleus aforethought, mammals got an earful of their ancestor's jaw" - J. Burns, Biograffiti jestingrabbit Factoids are just Datas that haven't grown up yet Posts: 5967 Joined: Tue Nov 28, 2006 9:50 pm UTC Location: Sydney ### Re: Mathematics and intellect antonfire wrote: t0rajir0u wrote: antonfire wrote:I generally find it more natural to read a series of formulae deriving some equation than to read a bunch English of sentences which do the same thing. I have read advice on mathematical writing which disagrees with this, and I personally find computations themselves highly unenlightening whichever way they're presented. (For example, I hate the computational proof of the orthogonality relations for finite groups. The conceptual proof is a million times clearer.) Anything you can do to make a computation seem like something other than mere algebra, in my opinion, helps. Perhaps, but I suspect you still prefer "the probability density at [imath]x[/imath] of a normally distributed random variable centered at [imath]x_0[/imath] with variance [imath]\sigma[/imath] is [imath]\frac{1}{\sqrt{2\pi\sigma}} e^{(x-x_0)^2/\sigma}[/imath]" to "the probability density, at some given position, of a normally distributed variable is one over the square root of two pi times the variance of the distribution, multiplied by the exponential evaluated at one over the variance times the square of the distance between where the distribution is centered and the given position". In this case (and in many other cases) I think the notation makes it denser and easier. Yeah, this is what people did before the invention of algebra. The idea that we should go back is madness. ameretrifle wrote:Magic space feudalism is therefore a viable idea. skeptical scientist closed-minded spiritualist Posts: 6142 Joined: Tue Nov 28, 2006 6:09 am UTC Location: San Francisco ### Re: Mathematics and intellect t0rajir0u wrote:Anyway, I should probably cite my sources here; Munkres' advice is here and Knuth's advice is here. Admittedly it is less extreme than I remember, but look in particular at the second section of Knuth's. Yeah, that doesn't seem to support what you were saying at all. Their advice is not to avoid notation, but to include both notation (when appropriate) and ordinary English (when appropriate) - nobody beyond the level of a second-year math undergrad should need to be told this, and indeed, the target audience seems to be something like a first- or second-year math/CS undergrad (maybe a little later for CS majors, since they have fewer classes with proofs). t0rajir0u wrote:I have read advice on mathematical writing which disagrees with this, and I personally find computations themselves highly unenlightening whichever way they're presented. (For example, I hate the computational proof of the orthogonality relations for finite groups. The conceptual proof is a million times clearer.) Anything you can do to make a computation seem like something other than mere algebra, in my opinion, helps. A but that's a matter of two different proofs, and which you find more enlightening, whereas the more/less notation question should be about the best way to present a given proof. I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side. "With math, all things are possible." —Rebecca Watson Dason Posts: 1311 Joined: Wed Dec 02, 2009 7:06 am UTC Location: ~/ ### Re: Mathematics and intellect antonfire wrote:Perhaps, but I suspect you still prefer "the probability density at [imath]x[/imath] of a normally distributed random variable centered at [imath]x_0[/imath] with variance [imath]\sigma[/imath] is [imath]\frac{1}{\sqrt{2\pi\sigma}} e^{(x-x_0)^2/\sigma}[/imath]" to "the probability density, at some given position, of a normally distributed variable is one over the square root of two pi times the variance of the distribution, multiplied by the exponential evaluated at one over the variance times the square of the distance between where the distribution is centered and the given position". I actually prefer when the pdf is actually a pdf. But that's just me. Correct me if I'm wrong but shouldn't it be $f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ where I'm using [imath]\mu[/imath] to denote the mean and [imath]\sigma^2[/imath] to denote the variance. double epsilon = -.0000001; mouseposture Posts: 42 Joined: Tue Sep 15, 2009 2:42 am UTC ### Re: Mathematics and intellect Mokele wrote: Yes, but for those of us without$40 million dollar surgical suites, it's not exactly a good idea to put a laptop through the autoclave.

For those of you without \$40 million surgical suites, it's not a good idea to do the kind of surgery that requires really scrupulous sterile technique . Not all of it does (you probably already know this, I realize).

Seriously, though, if you base your argument on the limits of current technology, then your conclusion will only remain valid until the technology improves. If you base your conclusions, as I do, on the limitations of the human mind, your conclusions will remain valid ... until the technology improves. Different technology, and the wait will be longer, but I seem to have argued myself into a corner, here.

t0rajir0u
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### Re: Mathematics and intellect

skeptical scientist wrote:Their advice is not to avoid notation, but to include both notation (when appropriate) and ordinary English (when appropriate) - nobody beyond the level of a second-year math undergrad should need to be told this, and indeed, the target audience seems to be something like a first- or second-year math/CS undergrad (maybe a little later for CS majors, since they have fewer classes with proofs).

Fair enough. We've been discussing this point for so long that I've forgotten what my original point was, which is this: non-specialists find specialized notation much more confusing than specialized terminology. Recall, for example, the old adage that every formula in a book cuts its sales by half.

achan1058
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### Re: Mathematics and intellect

skeptical scientist wrote:Yeah, that doesn't seem to support what you were saying at all. Their advice is not to avoid notation, but to include both notation (when appropriate) and ordinary English (when appropriate) - nobody beyond the level of a second-year math undergrad should need to be told this, and indeed, the target audience seems to be something like a first- or second-year math/CS undergrad (maybe a little later for CS majors, since they have fewer classes with proofs).
You wish...... They shouldn't need to be told that, but they often still do.

polymer
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### Re: Mathematics and intellect

Mokele wrote:
chilled wrote:erm.... biology versus anatomy...

What do you think anatomy is studying, rocks? Or maybe, oh, *living things*? You know, things with evolutionary and ecological contexts which govern anatomy, and which anatomy can elighten us about?

BlackSails wrote:Surgery isnt a kind of science.

It's called "making a point", using what is the equivalent of engineering for biology. The point being, you *need* these facts, lots of them, to even be able to function in biology, whether the issue is doing surgery on experimental animals (which, FYI, is a huge part of what we do), figuring out what's safe to grab in the middle of a jungle, or just being able to have an intelligent conversation with a colleague.

All science requires memorization; you memorize Maxwell's equations, I memorize the length-tension relationship of muscle, you memorize the periodic table, I memorize the phylogeny of vertebrates, you memorize how to do a Fourier transform, I memorize how to surgically implant electrodes. And it's not just random data - it's concepts that apply over huge ranges of life. Teach a student the Q10 rule and they have the core of understanding all thermal biology. Teach them the anatomy of a cat and they can find the kidneys in a shark, snake, frog and bird. Teach a student evolution and they can understand pretty much all life.

Just because it's not a formula doesn't mean it's not "a concept". And if physicists had to face the level of diversity we deal with, they'd quickly learn the merits of basic facts.

The problem is not that memorization isn't useful. It's the fact that it is a very really slow, energy consuming process, that doesn't significantly contribute to understanding. Most everything a physicist or mathematician is going to learn will be memorized through understanding of the concepts. It's almost a rule of thumb that anything actively memorized in math or physics is used as a crutch for an actual understanding of the subject matter. Thinking in terms of memorized equations instead of the concepts behind how some model or mechanism works is lazy and in the long run a serious handicap.

I'll concede that in a subject like Surgery, memorization is going to be useful if the knowledge being memorized is more or less arbitrary. It's understandable to memorize a network of veins if there is no clear justification as to why the veins take the paths they do.

However, if the mechanisms determining the structure of the veins were possible to understand, then simply memorizing where the veins went would be a serious handicap to whatever practice found that knowledge necessary, be it surgery or biology. It's okay to study a cat, learn all the components of how a cat works, because a good solid memory of the cats structure should follow from the disciplined understanding.

If a good solid knowledge of precisely how an organism is organized and looks is required, then the object should be memorized, but only when that knowledge is useful, since In my honest opinion the energy required(at least for me) to memorize something does not outweigh the benefits of actually trying to understand what's going on. In a field as diverse as biology, this point should be even further emphasized since it's strictly impossible to memorize absolutely everything.

So in response to you're argument regarding an environment, such as surgery, when dealing with unique organisms(Human anatomy should be memorized, that will be consistently useful), memorization should be the primary focus only once you know you're going to have to operate on said organism. This isn't always going to be possible, but it's a much more realistic and useful attitude then trying to memorize as many different organisms as possible.

BlackSails
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### Re: Mathematics and intellect

For all Feynman's making fun of biologists for memorization, he sure had alot of things memorized. Remember the story of how he told some engineers he could raise e to any power by running the taylor series in his head? He did it by having lots of memorized facts about different numbers.

njperrone
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### Re: Mathematics and intellect

Mokele wrote:
I love how physicists and mathematicians fail to understand the importance of facts.

Not so. I think you have just demonstrated very well a stereotype associated with mathematicians and physicist that helps categorize their intelligence.

I would find a professor very boring, and probably even a stand in for a real professor if all he did was look up formulas, methods of proofs, proofs, and terminology from the text for his course. Every profession has a unique basic set of facts that everyone memorizes for their job. The facts mathematicians require are different from that of physicists, which is different from chemists, which is different from biologist, etc.

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### Re: Mathematics and intellect

BlackSails wrote:For all Feynman's making fun of biologists for memorization, he sure had alot of things memorized. Remember the story of how he told some engineers he could raise e to any power by running the taylor series in his head? He did it by having lots of memorized facts about different numbers.

Yes, but I suspect he memorized a lot of facts because he used them a lot and had a good memory, not because he sat down with a list of mathematical and physical constants he wanted to memorize. If you use [imath]\sqrt{2}[/imath] enough in numerical applications, you start to remember that it's about 1.414, and similarly with any other constant that frequently crops up in equations, all without wasting any time memorizing things that can easily be looked up.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

BlackSails
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### Re: Mathematics and intellect

Well yeah, but I would guess thats how biology students memorize maps of cats as well. They look it up often enough, and they remember it

Mokele
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Location: Atlanta, GA

### Re: Mathematics and intellect

polymer wrote:However, if the mechanisms determining the structure of the veins were possible to understand, then simply memorizing where the veins went would be a serious handicap to whatever practice found that knowledge necessary, be it surgery or biology. It's okay to study a cat, learn all the components of how a cat works, because a good solid memory of the cats structure should follow from the disciplined understanding.

If a good solid knowledge of precisely how an organism is organized and looks is required, then the object should be memorized, but only when that knowledge is useful, since In my honest opinion the energy required(at least for me) to memorize something does not outweigh the benefits of actually trying to understand what's going on. In a field as diverse as biology, this point should be even further emphasized since it's strictly impossible to memorize absolutely everything.

So in response to you're argument regarding an environment, such as surgery, when dealing with unique organisms(Human anatomy should be memorized, that will be consistently useful), memorization should be the primary focus only once you know you're going to have to operate on said organism. This isn't always going to be possible, but it's a much more realistic and useful attitude then trying to memorize as many different organisms as possible.

I think the problem is that biology is not like physics - as a field of knowledge and understanding, it's fundamentally different.

In physics, you have a system which works according to invariable, immutable laws written into the universe itself. Everything is a product of those rules, and once you know the rules, you know the system. A photon is a photon is a photon, all the same no matter when or where. And there are no exception - the rules are inviolate. In a system like this, memorizing those concepts, those rules, is the only thing that makes sense.

In biology, you have a system which is subject to rules, but runs on pure, undiluted chaos (mutation). Some rules are truly universal (usually those based on physics, such as the drag on a flipper or the effects of temperature on chemical reactions), while others are the effect of a single, arbitrary choice made millions or billions of years ago and now rule an arbitrary subset of life (such as the five-fingered hand as the baseline and maximal limit for tetrapods, or 7 cervical vertebrae in mammals).

Here's the thing: both fields are attempting to maximally understand the system, to find rules as generalizable as possible. In physics, this is "easy" (from our perspective): you know the rules for some data, you don't for other data, so you try to mesh together the rules or uncover new ones using your data (acquired using the "easy" method of building a machine the size of a city). In biology, you're looking for a signal among a LOT of noise, and it can be *very* hard to parse out what's a real signal and what's an artifact of development, evolution, or just plain randomness. When we find an exception, does that mean the theory is wrong, or just that the organism is a legitimate exception that proves the rule? Add in the fact that we have extremely limited access to any step other than the final result (so many questions would be solved if we just had one live Allosaurus), and you can see why it's useful to have some level of encyclopedic knowledge. I've seen entire theories shot down with a single "trivia point" about some animal barely anyone even knows exists, and I've ruined complex mathematical reasoning with a single anecdote about what a particular animal did while trying to savagely maul me. After all, how can you formulate a general rule if you don't know enough about the system to spot exceptions?

Tl:dr - Physics is deterministic and rules-based, lacking variability or exceptions. Biology is literally fueled by pure chaos, resulting in a complicated mix of true general rules, arbitrary pseudo-rules (which may still be vitally important), bizarre variability, and vexing exceptions, all of which must be untangled in order to be understood, and failure to account for the most obscure animal or fossil may completely undermine a given attempt at more general understanding.
"With malleus aforethought, mammals got an earful of their ancestor's jaw" - J. Burns, Biograffiti

polymer
Posts: 125
Joined: Mon Feb 04, 2008 7:14 am UTC

### Re: Mathematics and intellect

That's very true, there are going to be many aspects about biology that don't follow a fundamental set of rules. My point was, was that most things work the way they do for a reason, and understanding why they work they way they do is probably more important then memorizing their outcome. There are going to be plenty of exceptions, and it is a bit of a trick remembering what they are, but even the exceptions have some underlying structure which can be learned. Memorization certainly has its place, I'd just personally take caution recommending it over understanding if understanding is possible. Especially since understanding usually leads to memorization anyways.

Mokele
Posts: 775
Joined: Fri Aug 21, 2009 8:18 pm UTC
Location: Atlanta, GA

### Re: Mathematics and intellect

My point was, was that most things work the way they do for a reason, and understanding why they work they way they do is probably more important then memorizing their outcome.

Yes and no. In some cases, the "reason" is that 350,000,000 years ago one salamander got eaten by a freshwater shark and it's mutant brother didn't - and that's why we have 5 fingers. Had the shark missed, we might have just as easily had 4 or 6 or 8 fingers on each hand (our early ancestors had tremendous variability in digit number, up to 23). An asteroid got bumped into a new orbit, and for no good reason the planet is over-run with over-evolved rats (mass extinctions have been shown to be essentially random). Some answers are just so far buried in the past that we'll never actually know without a time machine.

Plus, well, a lot of the time, we don't know why A leads to B,C, and D, but we do know that B,C and D affect X, Y, and Z, and so if we're studying Y, we need to know B, C and D.

There are going to be plenty of exceptions, and it is a bit of a trick remembering what they are, but even the exceptions have some underlying structure which can be learned.

Usually, but not always. Some lizards can lose and regenerate tails, others can't, and there's no rhyme or reason to the pattern. Some frogs have fused shoulder girdles, others don't, no pattern whatsoever. Ditto for sex determination in turtles. Sometimes who has a trait and who doesn't can be as simple as who has had the dumb luck for the mutation to occur in their lineage, resulting in pure, random scatter.

Simply put, sometimes things really are just random.

I'm not saying memorization is *better*, just that sometimes it really is necessary.
"With malleus aforethought, mammals got an earful of their ancestor's jaw" - J. Burns, Biograffiti

Woxor
Posts: 506
Joined: Mon May 07, 2007 11:28 pm UTC

### Re: Mathematics and intellect

DavCrav wrote:1) What is the second law of thermodynamics?
2) How many electrons in an atom of potassium?
3) How does a refridgerator work, i.e., cool things down?
4) What is a xylem?
5) What does it mean for a company to be highly geared?
6) What is bauxite?
7) What's the difference between mean and median?

1. Knew it.
2. Didn't know it without looking it up. Chemical engineer here. (Okay, I probably could have worked it out, but really, that's not what you need to know about potassium.)
3. Knew it.
4. Didn't know it, or forgot. Majored in biology for two years.
5. No idea.
6. Didn't know it.
7. Knew it.
8. No idea.

My suggestions for a better quiz:

1. What is the second law of thermodynamics?
2. Why does wood burn?
3. Why does a fan cool you down?
4. What is natural selection?
5. What effects do banks have on the economy?
6. Why does iron rust? Bonus: Why doesn't aluminum (seem to) rust?
7. What is calculus?
8. Describe the digestive system.

*shrug* As you can see, my preferences lean more towards testing concepts, which can include but are not limited to simple facts and vocabulary. Gauging scientific knowledge in particular requires this approach, IMO.

t0rajir0u
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Location: Cambridge, MA
Contact:

### Re: Mathematics and intellect

Agreed. The ultimate test of a scientific education should be whether you can apply it to make sense of the world around you. For example, any educated person should be able to explain things like how seasons work (my guess is that surprisingly few adults will be able to explain this correctly).

Qaanol
The Cheshirest Catamount
Posts: 3069
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### Re: Mathematics and intellect

Woxor wrote:5. What effects do banks have on the economy?

In theory, or recently?
wee free kings

DavCrav
Posts: 251
Joined: Tue Aug 12, 2008 3:04 pm UTC
Location: Oxford, UK

### Re: Mathematics and intellect

1. What is the second law of thermodynamics?
2. Why does wood burn?
3. Why does a fan cool you down?
4. What is natural selection?
5. What effects do banks have on the economy?
6. Why does iron rust? Bonus: Why doesn't aluminum (seem to) rust?
7. What is calculus?
8. Describe the digestive system.

But this wasn't the point! With a sufficient amount of time, you should be able to work out many things about the outside world. Luckily, people write this stuff down, so we don't each have to do our own experiments to find out the electropositivity rankings in the case of 6. (I don't see why 6,7 and 8 aren't just facts that have been memorized anyway. You can come up with the laws of thermodynamics given enough time, although you won't know which is the second one, of course. Number 4 is the ultimate example of reason being used to come up with a concept.)

In arts subjects, you cannot use logic and reason to find things out: no amount of thinking will lead you to know who Prokofiev was (and I hope everyone here does know who he was, or I am going to lose this argument with the arty people). My point was that while scientists tend to know a lot of arts and literature stuff, the converse is not true. Hence I just used some basic facts in science, taught to schoolchildren, as an example. I wasn't suggesting that arts people don't know how the world works itself (although this is also true) but that they lack even a basic knowledge of any science, and then scoff if you cannot name any works by Byron.

achan1058
Posts: 1783
Joined: Sun Nov 30, 2008 9:50 pm UTC

### Re: Mathematics and intellect

DavCrav wrote:In arts subjects, you cannot use logic and reason to find things out: no amount of thinking will lead you to know who Prokofiev was (and I hope everyone here does know who he was, or I am going to lose this argument with the arty people). My point was that while scientists tend to know a lot of arts and literature stuff, the converse is not true. Hence I just used some basic facts in science, taught to schoolchildren, as an example. I wasn't suggesting that arts people don't know how the world works itself (although this is also true) but that they lack even a basic knowledge of any science, and then scoff if you cannot name any works by Byron.
I am willing to bet that most people doesn't know Prokofiev. He isn't Beethoven. The only reason I know it is because I am big into 20th century classical music.