That's absurd. No college text is $20 new.
This seems like a good thread to ask the mathy people here if they've read Tobias Dantzig's Number and if so what they thought of it. I got a modern edition* cheap at a store formerly known as borders.
*So far as I know, it's the same text as classic versions, just a different layout/packaging.
It was apparently written similarly to Naked Came the Stranger. Each writer has finished a first draft of their assigned chapter of the proposed book to be titled "Mathematics through the Eyes of Faith." At Amazon you can do "Look Inside"--at a quick glance nothing really stood out beyond the premise itself.
And it seems to be part of a series.
Is mathematics discovered or invented, and why is it effective in the sciences?
No. Yes, you're stupid.
HarperOne seems to be a religion and spirituality imprint.
Q: What is the relationship between chance and divine providence?
A: He stopped at a drugstore to buy a Band-Aid.
That is why he is alive today..
(What a bunch of unregenerate Calvinists who are writing this inspirational spam!).
7: And that does not strike me as an unreasonable question for a college text. Apparentlly I am stupid.
7: Actually, I was curious about the discovered/invented distinction.
I've posted my answer in the archives somewhere.
Basically: neither. Math is a language. You use it to talk about ideas which are discovered, not invented. But a different language could be invented to discuss the underlying discoveries just as well.
Is mathematics discovered or invented, and why is it effective in the sciences?
It's not a bad question for a college course in isolation. But the context implies it's being used to bring religion in to the sciences, which sets off all kinds of alarm bells.
Ha ha stupid fundies, and I'm sure anything interesting Is handled very badly in this book, but there are a bunch of pretty interesting theological and philosophy of math questions around this topic. But oh ho ho stupid fundies.
10,11,12,13,14: I wasn't intending to answer either question, per se; I was intending to answer both questions with "no", because that's more succint than "shut up", and I was intending to cut off any follow-ups with "you're stupid", because I could see (as, I imagine, everybody else here could) where the follow-ups would lead.
Arguably, I should strive for greater clarity when speaking rhetorically to the unnamed authors of emails mentioned in posts.
I have no idea what the difference between discovering and inventing is.
Why math is useful in science is a very interesting question though.
16.last describes the point I was trying to make. Don't let them trick you, people who aren't horrible and stupid.
Why math is useful in science is a very interesting question though.
It is? I feel like I'm missing something.
I had this classmate, when I was doing my undergrad. He was very nice, and was in the same, relatively specialized, relatively computational major I was. We ended up paired up on a project, which was fine, although I think he wasn't actually terribly bright, all things considered. He was working on this movie the whole time, and at one point he had to beg off from some collaborative work to prepare for the premiere of this movie. "That's rad!", said I. "I would like to attend this movie. What is it?" "Oh, well" he said, "I dunno if it'd be your thing."
"Pshaw! I love movies, and you are my classmate, and you have made a goddamn whole movie as an undergrad! I would love to see it!"
"Well, okay."
The movie was called, I think, Supernatural (unrelated to the show), and it was obvious from even a cursory glance at the flyer that it was some freaky-ass christian shit.
Well, I didn't go.
Anyhow the next class after the premiere I said to my classmate "so, uh, how was your movie?"
"You know," he said "it was fine, but then at the end I started to talk to the audience members how if they believed what they'd seen, that all of these funny phenomena in the world were not scientifically explainable, they pretty much had just admitted that they believed in Jesus Christ, a lot of them up and walked out!"
Do not listen to or engage these people, unfoggetariat. Like Republicans, they are trying to TRICK YOU.
What about the guy with the three cards? He says I look like a winner.
Sorry about laughing at the fundies, Halford. To be fair, the post was posted without commentary.
It is? I feel like I'm missing something.
HOW IS IT POSSIBLE THAT MATHEMATICS, A PRODUCT OF HUMAN THOUGHT THAT IS INDEPENDENT OF EXPERIENCE, FITS SO EXCELLENTLY THE OBJECTS OF PHYSICAL REALITY?
23: for chrissakes don't apologize.
HOW IS IT POSSIBLE THAT MATHEMATICS, A PRODUCT OF HUMAN THOUGHT THAT IS INDEPENDENT OF EXPERIENCE, FITS SO EXCELLENTLY THE OBJECTS OF PHYSICAL REALITY?
Because both rely on sound logic?
Tobias Dantzig's Number
I very much enjoyed it (though, IIRC, it starts off a bit slow). I know I've recommended it before on unfogged.
I hope they at least give Wigner due credit for raising the question.
How do we really know what we know? Sure, scientists say they they are empirically testing belief, but isn't it true that "belief" when it comes to scientific theories doesn't exist if you don't have belief in something that transcends empiricism? Without belief in god, is there any belief in science? Food for thought.
UGH.
I fucking hate religion, really.
28: Yes, they mention him when it first comes up on page 8 and use his question to frame the section on it.
Math is a language.
Oh god heebie why did you have to go and say that what does it even mean arrrgh
31: what, math uses symbols to express concepts which map more or less directly on concepts in the real wo... OH SHIT THE SHARK IT'S EATING ME IT'S EATING ME
I dunno. It just doesn't seem very mysterious to me.
What we got here is... failure to communicate. Perhaps some ruminations on the metatheoretic foundations of our discourse are called for.
I know. That's my problem with philosophy...it always amounts to "why is that mysterious?" or "isn't the answer in the middle?" I'm really terrible at getting what's the big deal.
No, you're thinking of children's books.
34: Math is very useful for counting how many hardboiled eggs a man can eat.
35: Ludwig nailed in the preface to Tractatus:
On the other hand the truth of the thoughts communicated here seems to me unassailable and definitive. I am, therefore, of the opinion that the problems have in essentials been finally solved. And if I am not mistaken in this, then the value of this work secondly consists in the fact that it shows how little has been done when these problems have been solved.
Seriously, I have no idea what "math is a language" is meant to convey, even though people say it all the time. Similarly when people say things like "I'm fluent in 12 languages if programming languages count tee hee amn't I clever?" and the answer is no, shut up, shut up, please.
Sorry. I just... French is a language, you know? Cantonese is a language. Sign language is a language. Math? Just try translating this comment into math. No, really. I mean... what does it mean?
I do think it's pretty mysterious why the universe seems to operate in a way that can be expressed in terms of mathematics, especially mathematics that seems interesting and comprehensible to us.
I do think it's pretty mysterious why the universe seems to operate in a way that can be expressed in terms of mathematics, especially mathematics that seems interesting and comprehensible to us.
Because we're... good at explaining why things are comprehensible to us?
I mean, really good. Don't get me wrong. Damn good.
39: However:
For since beginning to occupy myself with philosophy again, sixteen years ago, I have been forced to recognize grave mistakes in what I wrote in that first book.
39: +it. 42: Later Ludwig in case it was not obvious.
It's really bizarre that repeatedly math which was investigated purely because it seemed interesting, later ended up appearing in the real world. This happened recently in my field, where the mathematical objects that I study turn out to describe certain systems in condensed matter physics.
It's really bizarre that repeatedly math which was investigated purely because it seemed interesting, later ended up appearing in the real world.
No it's not.
41 Because we're... good at explaining why things are comprehensible to us?
I'm not sure I understand what this means.
The questions I kind of want to ask are things like "why does the universe know what a Lie group is?" Not very well-posed questions, but still.
I was going to type more but on preview, 44 says it better.
When I say math is a language, I mean that it's just a bunch of symbolic conventions that we agree on to discuss underlying ideas. It's not a primary language like French, but that's because the content it sets out to talk about isn't everyday content.
You could imagine a world in which the rules of physics behaved more like say historical linguistics than like calculus. We don't live in a world like that, and that's interesting. Now it may be the case that there's nothing interesting to say about that, or no interesting reason behind it, but at the very least it's an interesting observation.
You could imagine a world in which the rules of physics behaved more like say historical linguistics than like calculus. We don't live in a world like that, and that's interesting.
Only very slightly.
45: Is so.
I mean, we evolved to deal with things happening on distance scales of roughly a meter. Why should the mental tools we developed for that stuff turn out to tell us how the universe works at distances a billion billion times smaller and larger than a meter? Shouldn't that kind of stuff have been way beyond our grasp, or even our interest? Even given that we live in a universe that operates accordingly to orderly rules, why did they turn out to be rules we can figure out? It's weird.
Alternatively, I might have just had too much to drink tonight. But, I think it is weird -- Wigner wasn't crazy.
Sign language is a language.
Nope, but sign languages are languages.
Seriously, I have no idea what "math is a language" is meant to convey
You might look into the difference between natural languages versus artificial and formal languages.
But math isn't an artificial or formal language. When we talk about math we talk in ordinary language. Some people do math in English, others do math in Russian, others do math in French. Math has jargon, but so does any other subject some community of people are interested in.
The wikipedia article on Wigner's paper has a nice quote from Gelfand which nicely points out that something interesting is happening:
"There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology"
It's also interesting that water happens to be denser as a liquid than as a solid; if it were the other way around, the environment wouldn't be particularly hospitable. Which I guess is to make the not very profound point that it takes a lot of exceptional circumstances to make our world habitable and I guess I'm not surprised that this seems to apply to what makes the world comprehensible. It's easy to see how this line of thought can shade into religion (though I'm not religious myself, and have never been).
It's really bizarre that repeatedly math which was investigated purely because it seemed interesting, later ended up appearing in the real world.
Imagine how much we don't understand because the math seems boring.
Just try translating this comment into math
math = math
I don't understand the quote in 54. And I think Tweety's position implies that if we understood different math, we would have figured out different physical phenomena, but maybe he's not comprehensible to me.
53 is the type of thing that prompts 35.
27: I actually think I first learned about the book here, or possibly through someone who comments/commented here but in comments on one of my old blogs.
But it's not like we have lots of mysterious physics sitting around which we don't understand waiting to be explained by boring math. We understand basically all physics which is accessible to our instruments. That's strange.
(There may very well be a counterargument of the form: "Physics is by definition the parts of science which are easily mathematicized." But I think that argument falls apart under more scrutiny. At the very least, because it's not *easy* math.)
61: Oh, I agree. I'm just trying to piece together a position from Tweety's dismissals.
Regarding your Wigner quote, wouldn't reductionism and essear's meter scale explain most of the effectiveness of math in biology?
regarding 54, there are a relatively small number of general relations that can be stated in mathematical terms that cover much of physics. Working out special cases and actually solving the equations takes a lot of work, but simple formal principles go a long, long way.
There is no mathematical definition of a species or an organism or a gene.
I don´t see how calling math a language helps much. Is chess a language? Other analogies are frowned on. A symbolic representation of something else, even one capable of recursion, is much much weaker than a language.
Hmm. Given the model of science that operates in physics, is it really a surprise that the physics we (agree that we) understand is the physics we have the ability to measure (i.e. that can be subjected to the tests that are seen as key to establishing whether we understand something)? I'm sure I'm missing something.
61. Equations of motion for this, please"
http://www.youtube.com/watch?v=4n5AfHYST6E
Our physical understanding of materials relevant to biology is terrible. There may be nothing interesting in mesoscale physics, but there keep being interesting bits that get turned up when people look at it; de Gennes and Kadanoff found lots of interesting stuff by asking interesting questions 9and being very capable at finding answers0
For most of human history we did not understand most physics at the scales which we were able to measure.
It's easy math if you compare it to what seems to be needed for biology, though.
We've never had actually incorporeal cognition; I'm not willing to assume physics isn't built into our reasoning. At the least, we should find out how cephalopods do math.
65. No, not at all. The hydrogen atom or liquid helium could be as complicated as gelatin or turbulence; the wave equation could be an exotic special case instead of being ubiquitous.
It;s not so much that math is useful in physics, it;s that a very few bits of simple math take you very, very far.
Some physics may be built into our reasoning, but that physics does not include general relativity.
We don't have an intuition for 4-dimensional space, instead we first develop an algebraic theory of 2-dimensional space which allows us to generalize to arbitrary dimensions. GR then says that you can understand gravity by treating time as another dimension and studying curved geometry on it.
To the extent that we have physical intuition here it's two dimensional and involves objects roughly the size of us. GR is purely 4 dimensional and involves objects way way bigger then us. There's just no way that our ability to reason evolved in order for us to be able to understand general relativity.
67: We diid have a good understanding of much of Newtonian physics. It's our explanations that were flawed.
Hey, much-smarter-than-me people, did we ever talk here about the SEC's proposal to require certain financial deals* to be described in Python?
The linked blog post is 18 months old; I'd be curious to know if anything ever came of the proposal.
From my naive perspective, it sounds like a good idea. Still vulnerable to an arms race of euphemisms, of course, but better than the legal thicket we have now. Or not?
*Transactions? I dunno. Not my field.
The time from Newton to the present is a very small chunk of human history. Even post-Newton I think there were usually lots of things well within the range of observation which we didn't understand (e.g. a good theory of electromagnetism took a while).
69: How complicated the explanation is a different question from whether we can come up with an explanation. Simple math vs. complicated math also seems like a different question.
Certainly I'm counting Newtonian physics as "physics we understand" and as an example of the unreasonable effectiveness of math in physics. I mean the orbits of giant objects zooming through outer space are given by exactly the shapes that you get by cutting a cone by a plane, WTF?
I meant that we had a good intuitive grasp of mechanics before Newton, but our folk physics were wrong. Thinking a bit more, though, I'm not even sure that first part is true.
On the historical question, I see your point. What I guess I'm saying is that it's not a surprise that we've made the most progress in the areas where we can measure things, as those are the areas where progress is possible (with respect to our current ability to measure things). I'm not sure how we'd make progress in areas we couldn't measure. But yes, you're right that we could also simply not have made progress in the areas we can measure.
I think a hardcore constructionist would probably come in at this point and argue some other points that I don't really want to get into.
66, fair enough, I was overstating things. Though it's worth pointing out that the equations of motion for the center of mass of that cube of gelatin are probably quite simple.
70.3: no, but GR, our brains, & our environment developed from the same origins. Uplifted cephalopods would, I am imagining, have as much work to get to GR; but I'd really like to know if they worked out everything else in approximately the same order.
77: Right, the point was why we understand so much, I took for granted that we don't understand things which are beyond our scales of measurements.
You'd expect that animals that live in the ocean would probably be way less interested in astronomy than animals that live on land. Furthermore they'd be way less interested in gravity. So you'd definitely expect them to make discoveries in different orders, but ultimately things like 2-dimensional geometry are sufficiently simple that I'd expect any sufficiently advanced civilization to end up with enough geometry to end up with roughly the same version of GR eventually.
80: See, to me I see it as the explanatorily boring (and probably philosophically naive) result of having put so much work into it. Work isn't guaranteed to pay off, but once we reached a point where it seemed like we were on the edge of understanding the most that we could understand,* then we'd always be on that edge even as we kept pushing it along.**
*I'm thinking of the view - which could very well have been a minority position - about a century ago that we were running out of things to discover and might have to move on to other pursuits.
That second footnote was deleted, but I forgot the reference.
No, guys, Tweety says some major philosophical problem is stupid, so why talk about it at all? He's a DJ with a good sense of humor.
OK, 84 was dickish but come on. I love Tweety but that was straight bullshit.
I'm not sure that geometry is necessarily the only route to GR. You could imagine a world where it went: first get quantum mechanics and special relativity, then work out what consistent scattering amplitudes for spin-2 particles look like, then afterwards attach all the differential geometry interpretations to the formulas you find. A world where quantum mechanics and special relativity were understood before differential geometry doesn't seem all that crazy.
72. I would not expect technology to be able to solve a social problem. Maybe a step forward, but in my mind the real problem is that leveraged risk is profitable and easily tweaked to be just outside the reach of this year§s laws. Not my field either, I§m just a spectator
I think "unreasonable effectiveness" refers to the great power of simple descriptions in physics but not elsewhere. Who cares whether in principle you can describe something if the resulting math is too complicated.
Lord Kelvin said everything had been discovered in 1900. Michelson as well. Kelvin was brilliant and extremely productiveů he estimated the age of the earth from basic thermodynamic principles. He didn§t know about radiative decay, so was off by a factor of 200,
87.3: A great Kelvin/Rutherford anecdote. Rutherford was nervous because part of his talk would highlight how far off Kelvin had been and Kelvin was in the audience.
To my relief, Kelvin fell fast asleep, but as I came to the important point, I saw the old bird sit up, open an eye, and cock a baleful glance at me! Then a sudden inspiration came, and I said Lord Kelvin had limited the age of the Earth, provided no new source [of heat] was discovered. That prophetic utterance referred to what we are now considering tonight, radium! Behold! The old boy beamed upon me.
On the general discussion, I do not trust myself to navigate the various fallacies that lurk in that parts of the discussion which hinge upon the anthropic principle. But still, nothing to be sniffed at, apes that got past some threshold of complex processing, and voila! Here we are.
81, 86: I don't know that cephalopods would need Euclidean geometry at all; I can imagine them doing chemistry before mechanics, & following essear's order because of that.
And if the kind of body matters, embodiment might matter.
After all, cuttlefish write in sepia ink.
I'm a straight-up platonist and think numbers are as real as animals in the zoo.
not a completely rare position among the mathematicians I knew in grad school. of course, one of them was crazy (brilliant, but headed for schizophrenia), but there you go. and frankly the set theory guy was a little off as well. nonetheless, I'll stick to my guns on it. additionally, I find this explanation to be more or less self-evident, such that I really don't understand explanations like sifu's or heebie's at all. well not, not understand, more like I think, what on earth could have led them to that crazy conclusion, when the idea that numbers are real and exist independent of humans to think about them is so obviously right? just like when I had a friend who said geometric shapes don't exist independently of people to invent them and I said, dude, you know there's crystals forming everywhere all over the world? what are they but hexagonal and so forth?
I do think it's pretty mysterious why the universe seems to operate in a way that can be expressed in terms of mathematics, especially mathematics that seems interesting and comprehensible to us.
Only because you're looking at it the wrong way. The mathematics that describe this universe are an inherent aspect of the universe and worrying about how it;s possible that they can describe so well the universe they're a part off is as pointless as wondering why our concept of a blue sky describes so well what we see outside on a clear sunny day.
Everything else in this vein is just us thinking how darned special we are and how lucky we are to live in a universe just right for us, putting consequence in front of cause, or descartes in front of the horse.
Philosophy is all very well, but the important thing to take note of at this hour in the morning is that smoke alarms that go off to tell you that their batteries are running down are fucking annoying. Four thirty in the goddam morning, and I wake up to "Low... battery... (long pause) low... battery... (repeat ad lib)" And I already hadn't been sleeping well because Buck is out of town. Feh.
Very Wisse. I can't help finding all the arguments that take the rough form of the anthropic principle to be kinda unsatisfying, like wondering how I came to be just the right height and width to fit through a standard door frame. Is a bit of intellectual humility really that difficult?
Meanwhile, Pythonic finance. There's an XML dialect for financial reporting (XBRL), so the next step is obviously to set up an application based at the SEC that scrapes all the XBRL disclosure files, executes the python code, and verifies the consistency of the accounts.
In the event of a discrepancy bigger than a given amount, it pushes the location of head office into the targeting interface of a squadron of MQ-9 Reaper drones assigned for this particular task. I feel this would be a real brake on the rate of retrogression in the quality of public accounting.
Obviously you'd need to have some pretty good precautions against code-injection attacks or some bright boy would try it on.
It's not clear that there should be one single answer as to why mathematics is so effective. Maybe there are different answers in different situations. For example, the effectiveness of statistics is explained by things like the law of large numbers and the central limit theorem, but other areas need other explanations.
Math may not be a language, but back when I was in grad school I taught the intro calculus one year. I discovered half way through the sememster that the top student in the class spoke very little English. I asked her how she followed the lectures or read the book. She said she just looked at the formulas.
proposal to require certain financial deals* to be described in Python?
"I wish to lodge a complaint about this company I have bought. It's bankrupt."
"Oh, no sir. It's just... resting."
"Resting?"
She said she just looked at the formulas.
Voila the script in which the language is written.
I can't help finding all the arguments that take the rough form of the anthropic principle to be kinda unsatisfying, like wondering how I came to be just the right height and width to fit through a standard door frame.
"It is demonstrable," said he, "that things cannot be otherwise than as they are; for all being created for an end, all is necessarily for the best end. Observe, that the nose has been formed to bear spectacles--thus we have spectacles. Legs are visibly designed for stockings--and we have stockings. Stones were made to be hewn, and to construct castles--therefore my lord has a magnificent castle; for the greatest baron in the province ought to be the best lodged. Pigs were made to be eaten--therefore we eat pork all the year round. Consequently they who assert that all is well have said a foolish thing, they should have said all is for the best."
You'd expect that animals that live in the ocean would probably be way less interested in astronomy than animals that live on land.
I don't know about that. Wouldn't it be rather important for them to know about tides and seasons? And long-distance navigation in the absence of landmarks is a lot easier when you look at the stars.
Of course, tool use - and therefore constructing telescopes - would be rather trickier if you lived in water. Maybe the heavens are full of worlds inhabited by intelligent but non-technological sea creatures, gazing up at the stars uncomprehending forever.
Hey, much-smarter-than-me people, did we ever talk here about the SEC's proposal to require certain financial deals* to be described in Python?
The linked blog post is 18 months old; I'd be curious to know if anything ever came of the proposal.
It's in force, as far as I know.But it's not the deals as a whole that are described in Python, but specifically the cashflow waterfalls. Which is basically a series of if...then tests determining how principal and interest receipts on the underlying assets should be distributed to the different investors in a securitisation. So, eminently suited to being described in programming language. Previously, they were described in legalese in the transaction documents, and various third parties would attempt to translate that into code so that the cashflows could be modelled. The rule standardises this process and assigns responsibility to the issuer (technically the depositor, I think). That way everyone is using the same, official waterfall and they don't have to pay for it (the third party providers charge a lot of money).
99: The Unreasonable Effectiveness of Python in the Online Enterprise.
Meanwhile, Pythonic finance. There's an XML dialect for financial reporting (XBRL), so the next step is obviously to set up an application based at the SEC that scrapes all the XBRL disclosure files, executes the python code, and verifies the consistency of the accounts.
Accounts can be consistent but still wrong. Or rather not representing a true and fair picture of the company's finances. That doesn't mean there aren't red flags that a script could pick up (eg large discrepancies between certain quarter end figures and the averages), but consitency alone doesn't help much with modern day fraud or malfeasance, at least for listed companies.
but consi[s]tency alone doesn't help much with modern day fraud or malfeasance, at least for listed companies.
See also Sarbanes-Oxley Compliance.
Holy crap do I not understand this thread.
It's also interesting that water happens to be denser as a liquid than as a solid; if it were the other way around, the environment wouldn't be particularly hospitable.
Because your drink would only be cold on the bottom. To avoid a warm drink, I suppose we'd have to use straws and no adult should drink from a straw in public unless they've had their jaws wired shut or something.
I'm glad 50 took my comments in spirit of light-hearted belligerence in which they were intended.
I'm also fairly pleased that, despite my expected (and desired!) total failure to ever stop anybody from talking about anything, I apparently have an uncanny ability to get under halford's skin. Hey halford: GM cars are junk!
Anyhow the real question underlying Commenter and essear's take seems to me to be "why did we develop analogical reasoning?" because that seems to be the core of all this, unless you want to back it out to "why are things like other things?" which is I'm sure a very deep philosophical question but is still not really germane, because it's definitely how things are in this universe and this universe is the one we've been looking at, in general.
I spent a couple of hours this week listening to somebody try and convince me that the theories of cognition that (more-or-less) underlie this book are fallacious, but they were unsuccessful in that effort.
Hey halford: GM cars are junk!
My dad stopped GM after 40 years of it. That last Buick was too much trouble. On the other hand, I've always owned Chrysler products. I think that disproves natural section.
"why did we develop analogical reasoning?"
Are you arguing that math is an analogy?
Not that it is an analogy, but that in some important and illuminating ways it is like an analogy.
Not all of it, no. Obviously, like, subitizing isn't analogical. But our ability to figure out how things work on scales other than a meter is definitely a function of our ability to reason by analogy.
109.last: No, it's why our Plains Ape analogies are unreasonably effective.
"Math is an analogy" is an analogy.
Much as I'm enjoying the implications of math being an analogy, I should probably specify that, no, I'm not arguing that math is an analogy. I'm arguing that lots (most?) of math was developed by thinking about way things are analogoous to other things.
115: prior supposition of unreasonability does not a mystery make. People have been trying to improve and refine and generalize and formalize math for, what, tens of thousands of years?
Going back to the discovered/invented question, I've been thinking out this. I have decided:
Basic arithmetic - discovered
Algebra - invented
Geometry - discovered
Trigonometry - discovered
Calculus - invented
Set theory - I have no idea what this means
Statistics - invented
. On the other hand, I've always owned Chrysler products. I think that disproves natural section.
Not at all, Moby, it's handicap theory. Females are impressed by your driving a Chrysler on the grounds that, if you can still get where you are going even in one of those, you must be a terrific driver. Like a peacock's tail, the message of which is "hey, I'm so strong and fit that I can manage to fly and survive even with this ridiculous thing attached to my bottom".
I don´t see how calling math a language helps much. Is chess a language? Other analogies are frowned on. A symbolic representation of something else, even one capable of recursion, is much much weaker than a language.
From way above, this is the type of thing I find totally exasperating. It's not an analogy. It may be imprecise to call it a language, but we all know what we're talking about: symbols and conventions for communicating content.
124: How does that way of thinking separate "math" from "notation"?
You* can write calc stuff in at least two different ways.
*I can't, but I'm assuming you can.
I'm off to stats class, where I won't understand the derivations because I don't speak calc.
The language in 119 has problems. The math is fine.
124: How does that way of thinking separate "math" from "notation"?
I'd loosely be okay calling these synonyms. Notation + conventions + rigor = math.
Anyhow, I don't really get the "math is a language" thing, I must admit. Are languages axiomatic?
"Math is language" just means that the person is talking about the scribbles and symbols and graphite on the paper, not platonic ideals of THE NUMBER FOUR! that exist in a bubble near the ceiling of the classroom.
EVERYONE GOT IT?
symbols and conventions for communicating content.
But when people want to know whether math is discovered or invented, what they're asking about is the content: i.e, are the truths of mathematics independent of, for example, contingent psychological facts about how humans reason or not?--when we make "discoveries" in mathematics are we doing something more than simply extrapolating the consequences of axioms and rules of inference that we've agreed are legitimate?
I'm not seeing how calling mathematics a "language" really gets any traction on those questions.
Heebie's a notationician, not an arithmetician.
Because it's a trick question. Parts of math are invented, parts are discovered. Calling it a language identifies the part that is invented - the part specific to that group of people - they invented the notation needed to talk about the ideas. Calling the actual ideas out there "discovered" seems off to me, but I mostly just can't get worked up about it either way.
132: If there isn't some real "4" that exists outside various written versions of it, then how come I have four fingers on each hand?
135: it's not a trick question; the question of whether there are "ideas out there" is the whole meat of it. You're attempting to side-step it but unsuccessfully; nobody in the history of ever has argued that mathematical notation was discovered in its platonic form.
136: you lost your thumbs in an accident?
I didn't make the joke explicit. I discovered the joke.
This classroom has a motorized blackboard.
Why don't people ever talk about the discovery of, like, verbs? Why should it be that verbs do such an extraordinary job of communicating the relationships between objects?
the question of whether there are "ideas out there" is the whole meat of it.
Then this is a question that I basically don't get. (Off to teach or I'd elaborate.)
141: People spend a great deal of time talking about the development of language and how much of it is innate and why certain concepts keep reoccurring.
143: oh boy do I know it. But still, nobody talks about humans discovering linguistic concepts that were out there; they talk about the human faculty of language evolving to suit the purposes for which it was most useful.
You could certainly talk about whether mathematical ideas are innate or not; some clearly are, and there's a fairly tedious amount of disagreement about others. But that's different from talking about whether math exists independent of humans thinking about/caring about math.
Right, but nobody doubts there is a "tree" independent of the word.
146: I respect your ability to reason analogically, but reject your analogy.
Surprisingly on point: My son's third grade math worksheet (from a major publisher) had the following question on it last night.
"The area of Israel is 8,550 square miles. Write the area of Israel in word form."
WTF? Isn't this question kind of sensitive for a third grade math homework assignment? A fair number of American third graders have family members who died in the many wars and terrorist attacks about the size of Israel.
Also, it seems to be wrong, according to the United States of America. The CIA factbook and Wikipedia say 20,700 square kilometers, which Wikipedia translates to 7,992 square miles. Infoplease.com says 7,849. Answers.com says 8,367 sq mi. look-israel.com says Israel has 8,463. Various other sites have numbers either just under 8,000 or in the 83-8500 range. I'm pretty sure that the difference is a political issue, maybe inclusion of the West Bank.
Clearly a Zionist conspiracy controls American textbooks.
It's only on the rare occasion that I get outside of light pollution and actually see the real sky that humans saw prior to the industrial revolution that I realize why people cared so much about astronomy so early on. Yes "intelligent cuttlefish" would understand tides relatively early on, and they might get a theory relating that to the moon. But there's no way in hell that they'd discover planets anywhere near as early as humans.
150: What is the answer supposed to be? Is it "Eight thousand five hundred fifty square miles..."?
I can't determine if this indicates that I need to repeat 3rd grade, or that you should be home schooling your son.
152: you forget "and". That's okay, third grade is nice.
151: Yeah, the answer was to write out the number in words. Aside from politics, it's a useful exercise for understanding where to say "thousand" etc. Rely on other evidence ot determine if you should repeat third grade.
Integers look Platonic when you think about them, but objects invented to prove a point, say the Banach-Tarski paradoc or Kuramoto-Sivashinski equation are as artificial as Rubik's cube. Where is the boundary between nature and artifice?
We are not the first to consider this question. Two essays that I like, both of which stress the importance of broader context for mathematical ideas, are Quine's Two Dogmas of Empricism
and Kac and Ulam's Mathematics and Logic,
one good chapter on Google books.
151: planets are just wandering stars, they don't need any particular technology to discover.
We are sort of assuming an intelligent sea creature with eyes that work out of the water. Not all sea creatures have such eyes. Dolphins and seals do; others don't. An intelligent creature that couldn't see in air would have serious problems seeing the stars and even the moon.
153: I didn't forget it. I made a conscious decision that it was not needed.
I guess that settles it. The school year has already started, but maybe if I beg they'll let me in anyway.
We are not the first to consider this question
What? No!
Now I am off to stats class, where I will be enraged by the stupidity of the way it is taught.
I was not assuming intelligent sea creatures that could see out of the water. (After all, it's not like our intelligence allows *us* to see very well under water!) Nonetheless I'd bet they get to the moon pretty quickly. The full moon is really bright, it should be noticeable near the surface under water.
As far as I know, cephalopods are unique for having intelligence (certainly an ability to manage spatial relations, possibly more) coupled with a short lifespan. Octopus live only a few years, reproduction kills them.
Intelligent birds, cetaceans, and other primates all have much longer lifespans. Finches sometimes get cited, they have complex songs, but not nearly as capable of learning as are parrots.
Now I am off to stats class, where I will be enraged by the stupidity of the way it is taught.
I am currently taking an online class on UNIX (part of a programming certification) in which the only work that is actually graded are multiple choice questions from some obviously textbook-provided "instructional technology," e.g.:
"Which of the following is not a commonly used text editor in UNIX?"
a. . . .
b. . . .
c. edlin
d. . . .
e. . . .
Because being able to recall the names of text editors (rather than, say, actually using one of them to create shell scripts to solve a problem) is exactly the kind of skill programmers need to master.
To say the instructional quality in this program varies greatly by instructor doesn't even begin to describe, etc.
As far as I know, cephalopods are unique for having intelligence (certainly an ability to manage spatial relations, possibly more) coupled with a short lifespan.
This is probably a good thing. Imagine what the evil tempered little sods would pull off if they had seventy years to plot in.
Then this is a question that I basically don't get. (Off to teach or I'd elaborate.)
Back from teaching. Of course they're not out there in some fruity sense. "Blue" and "4" and "Lie Groups" are adjectives in our head. Sometimes they match up to something out there. Sometimes they don't, because we have imaginations.
Imagine what the evil tempered little sods would pull off if they had seventy years to plot in.
"Lie group" is an adjective? Looks like I don't understand math, philosophy, or grammar.
Among the rain
and lights
I saw the figure 5
in gold
out there
Squid and sharks already migrate vast distances without using the stars (probably ). If they became intelligent, they could start with chemistry, topology, fluid dynamics; metric spaces would be the late formalization.
165: Ha, Ginger Yellow, I was wearing one of those yesterday. Bit of a man-magnet, too.
Somehow I'm not surprised that there are a number of Google hits for "math is a verb".
All their scientific knowledge is held by a group of long-lived, celibate, cephalopod monks.
"Lie group" is an adjective? Looks like I don't understand math, philosophy, or grammar.
I regretted using "adjective" immediately after posting. I just meant "description".
168: they could start with chemistry, topology, fluid dynamics
Yes, they'd probably have subject called gradientology.
"If an intelligent alien squid were to design a school curriculum, what subjects would it include?"
If they became intelligent, they could start with chemistry, topology, fluid dynamics; metric spaces would be the late formalization.
Intelligent sharks would probably have worked out Maxwell's equations before Maxwell.
Well, somewhere in the corpus there'd be something analagous to Moby Dick.
175: Intelligent demon sharks.
(Don't worry the important recommendation is due in at 1:30, so this is time-limited.)
Why math is useful in science is a very interesting question though
This was the topic of my dissertation. I wanted to say that the question was badly put, and I had a better way of putting it, but now I just admit that I don't fucking know anything.
159: I enjoyed my stats class. Maybe you're just too easily enraged?
40 etc. I used to get stabby when I would read of the (Lacanian? I can't even remember, because it's been a long time since school and I'm getting rapidly stupider) idea that "the unconscious is structured like a language" because it's so goddamn meaningless. The best parallel I could come up with was "my kidney is structured like a foxtrot."
I am sorry I missed this thread until it was impossible to do anything but skim or else abandon all work for the day.
172 I regretted using "adjective" immediately after posting. I just meant "description".
But I don't think "Lie group" is descriptive in the same way "blue" is. "Blue" is a fuzzy human concept -- it doesn't have very well-defined boundaries, our idea of whether something is blue depends on lighting and our perception and different people might disagree at the edges. But "Lie group" is an exact concept. It's really exactly true that gluons interact according to the su(3) Lie algebra, and when I say "exact" I do mean it in a sense I can make precise. It's like energy conservation in that I believe it far more than I believe individual experiments and measuring apparatuses; screw it up, and things change dramatically. So it really is exact.
[Yes, energy conservation is somewhat mangled in general relativity, but again, there is an exact statement that I could make precise if anyone wants to force me to. I do think these are things we know about our universe that no amount of future information can ever change, and they happen to fit with mathematical structures people invented for other reasons. It's... weird.]
What's so aggravating about this "math is a language" phrase? We've got ideas in our head. We describe them with symbols. We call it math if it's rigorous.
But "Lie group" is an exact concept.
Because that's how mathematical content works. The idea has to be well-defined if you're playing by the conventions of math. "Four" is a well-defined, exact concept that functions very closely to "blue".
We call it math if it's rigorous.
What does saying "math is a language" add to that? The whole question becomes what is "rigor" and language is just a trivial side issue.
You're saying the discovered/invented riddle amounts to asking "what is mathematical rigor?"?
Because, easy: mathematical rigor is like jazz.
184: But why does nature like to play with the same mathematical concepts we do? Why does something some Norwegian dude dreamed up a century ago turn out to be how gluons interact when no one knew what the hell a gluon was for several decades after that?
But that's just interesting because Lie Groups and gluons are right on the boundaries of our knowledge. It's not very interesting that "4" describes Moby's fingers really well.
Hmm. Maybe it's more like obscenity then.
186: No. I'm saying what I find so aggravating about this "math is a language" phrase.
It's not very interesting that "4" describes Moby's fingers really well.
That's what the lawyer for the people who made the coffee grinder kept saying.
188: The way I've always thought of it is that mathematics all comes down to logical deduction. You start with the simple stuff, and keep on with using logic to prove more complicated things, and then you get to all the crazy math that I will never understand. And we've evolved to use logic (among other strategies) to understand the physical world because it does describe how physical objects behave on a gross level.
So it makes sense that the logic that evolved to help us deal with an observed universe that behaves logically continues to be applicable even in more esoteric areas that are not directly observable.
To say it another way, I have no idea what either a Lie group is, or what gluons are. But whatever gluons are, they have to behave in some kind of logically consistent way, because things in the physical world do. And any kind of logically consistent way that gluons could behave is going to be describable by some kind of math.
But it doesn't even stop there. As Commenter said, there are other mathematical structures that crop up all over the place, and some of them I think can even have similar "exactness" statements made about them (handwaving about universality goes here). It isn't weird that, like, particular lumps of material turn out to obey, in very precise senses, all sorts of rules that people made up just for the fun of it?
Somewhat surprisingly, I actually suspect (and I'm not a philosopher of Math) that the book we were all supposed to be mocking in the OP has more rigorous and accurate analysis of the questions under discussion than what we are seeing here.
The very first paper I wrote as a college freshman was on this topic. I remember citing a quote from someone to the effect of "mathematicians are constructivists in the workplace, but when they go home in the evening, they're platonists."
Update to 198: I understand why the question might be uninteresting, but I don't understand how saying "Math is a language" is supposed to related to that either.
199: or maybe it was formalists at work rather than constructivitists. It was a long time ago.
I think y'all would find these questions considerably more tractable if you just accepted Jesus as your personal lord and savior.
when they go home in the evening, they're platonists.
NTTIAWWT.
195: All deduction starts with a prior set of premises. These must either be postulated out of somebody's head (invented) or induced from observation (discovered).
Moby, I'm lost. There's some question that's interesting to you and others, and I don't get which part of it isn't obvious.
Two of the original questions seem to be getting run together.
The invention/discovery question is separate from the question of why the physical world is describable in terms of something that is invented/discovered independently of our experience of the physical world.
On the first question, I again don't see how
"playing by the conventions of math"
gets any traction on the issue. The question becomes what contrains the conventions. The conventions of chess, for example, are constrained by human agreement about what makes up the game. That's the clear sense in which chess is invented by human beings. Numbers, their properties, and their relations are clearly not like that.
206: I'm kind of a closet Platonist when it comes to math.
208 would probably be less confusing if I hadn't made a stupid joke on the same word in 203.
Almost all mathematicians not working on foundations are deep down platonists about the natural numbers. But for anything else (say real numbers or sets) there's no consensus.
I'm a bit of a Platonist about higher level concepts like the normal curve. Perhaps such thinking stems from my being religious or perhaps it is the statistical training with its constant references to unobserved (and unobservable) parameters.
197: Somewhat surprisingly, I actually suspect ... that the book we were all supposed to be mocking in the OP has more rigorous and accurate analysis of the questions under discussion than what we are seeing here.
Hmm, math textbook written by actual math professors vs. blog comments section. Surprising!
The way I've always thought of it is that mathematics all comes down to logical deduction.
YOU PEOPLE ARE FUCKING KILLING ME HERE. KILLING ME.
If the book just deals with questions like this, then I'm not clear what makes the book in the tradition of Christianity, rather than just the philosophy of math.
I'm not clear what makes the book in the tradition of Christianity, rather than just the philosophy of math.
The publishers' impression of their likely market?
214: I imagine most modern philosophers of math don't try to use the platonist position as an existence proof for jesus?
197: did you have specific problems with this discussion, or are you just confused by it generally?
We've got ideas in our head. We describe them with symbols.
This is a very strongly expressed view of language, that ideas always exist aside from their expression. There are people who disagree about the scope of this characterization, and cases where this does not apply (commands and exclamations, for examples in one direction, computer-generated theorems in the other). This disagreement is about scope is why idintifying math as a language means wildly different things to different people.
Some of the relations about numbers pretty clearly are invented by people; we can speak with great precision about integers that exceed the magnitude of the universe.
TBH and IMO, a basis for the philosophy of probability in divine providence wouldn't be that much less satisfying than any other currently running candidate.
oops, "disagreement about scope is why" is what I meant to write.
217.2: Without specifying the units, even?
We've got ideas in our head. We describe them with symbols.
MAKE THEM STOP, PLEASE KURT MAKE THEM STOP. WE CAN TALK ABOUT WHETHER "LOGICAL DEDUCTION" MEANS "AXIOMATIC SYSTEMS" LATER.
we can speak with great precision about integers that exceed the magnitude of the universe
I know the answer is going to make baby David Hilbert cry, but what do you mean by "an integer which exceeds the magnitude of the universe"?
???
application of measure theory and a particular set of combinatoric results.
Universe in units of the Planck length, say. People discuss properties of natural numbers too large and too small to be imagined, let alone referring to anything. My point is that in thinking about, say, partitions of integers, any connection with observable reality is quickly left behind.
Don't know if I'll ever read Godel, et al, but there was some Hofstadter thing that I read about analogy and cognition that was pretty interesting and possibly even relevant to this thread.
Personally I (and iirc, Wittgenstein) think that "mathematics is like a language" is about right in terms of being the best five word slogan you could have, but I think one has to admit that it would be a lot more useful an explanation if anyone anywhere had a remotely straightforward answer to the question "What is a language?" or indeed "what is the relationship between language and the real world?".
I have now joined the winning team and thus believe that academic philosophy made loads and loads of progress in the last fifty years, but unfortunately those two particular questions have remained a bit stubborn.
The Nagel and Newman book on Godel's proof is great. Hofstadter is just OK on this topic.
Universe in units of the Planck length, say.
why the Planck length, rather than (for any integer N), the Planck length raised to the power -N?
Because Budweiser is pisswater, that's why.
229: I recall somebody using the number of atoms or particles in the universe for that purpose. But, whatever the maximum number is, you can always multiply it by itself and get a bigger one.
Wittgenstein
You mean Tractatus WIttgenstein rather than Blue+Brown books W, right? Blue+Brown W attacked and soundly defeated Tractatus W with a fireplace poker. In front of witnesses.
I recall somebody using the number of atoms or particles in the universe for that purpose
Why the number of atoms in the universe, rather than the number of sets of atoms in the universe, or sets of sets of atoms in the universe, etc etc?
I MEAN THE WITTGENSTEIN OF "REMARKS ON THE FOUNDATIONS OF MATHEMATICS". I GUESS YOU MIGHT NOT HAVE HEARD OF THEM. THEY'RE PRETTY OBSCURE.
NO, NOT "LECTURES ON THE FOUNDATIONS OF MATHEMATICS". I ONLY REALLY LIKE THE POSTHUMOUS STUFF.
And then the Remarks on the Foundations of Mathematics Wittgenstein destroyed both of them with a folding chair.
233: I have no idea, but you can call the number of sets of sets of atoms in the universe X and then write down X^100 and you are discussing a number too large to be of any physical relevance.
236 was me, and posted before I saw 234.
235: they're denumerable. I have just got Baby Georg Cantor to sleep, please don't make him cry.
they're denumerable
It is almost impossible to denumber a car these days. The VIN is in so many places.
you can call the number of sets of sets of atoms in the universe X and then write down X^100 and you are discussing a number too large to be of any physical relevance
it's got a perfectly clear physical relevance; it's the number of sets of sets of sets of ... (iterated some number of times, can't be bothered to work it out) atoms in the universe. If you're going to allow such things as sets of things, you can get integers as high as you like by counting your feet (binary!)
I should add that I sort of see where you're going with this - the natural numbers are defined with reference to the activity of counting, rather than specific physical objects - but one shouldn't underestimate how many ways there are to get the simplest physical objects into the same relationships as mathematical entities, something which apparently puzzles a lot of people why that should be so.
241: Not if you want physical relevance. You need to keep a set to describe the dimensions you used to define your sets. Unless you are saying that your sets are somehow held in some state that is not part of the universe.
Not if Jesus counts your feet for you!
the natural numbers are defined with reference to the activity of counting, rather than specific physical objects
Right, if you are counting such that each set (where a set can be an individual item, a group of items, or a group of previously defined groups or whatever) is only counted once, you need a way to record what you have counted. If what you are counting is the smallest discernable unit in the universe, every set you define requires X number of units to mark it as counted and X is greater than one.
Not if Jesus counts your feet for you!
Or, you can ignore what I wrote above if somehow you have defined the universe to exclude whatever you are recording the count on
Wait, this is a serious conversation?
Large integers are just one example of naively Platonic mathematical objects whose boundary with invented constructions cannot be drawn.
If integers, then what about fractions? Continued fractions? Generalized continued fractions? Fibonacci numbers?
denumerable
Set of all sets that do not contain themselves as members, included in set of sets or not?
they're denumerable
Is this from a lost verse of "It's De-Lovely"?
246: Not, because there is no such entity, therefore it can't be included in any sets.
175, 176: Also Abbey Roadoe.
246: I assumed we had unwittingly wandered back to the post title.
Starting with natural numbers being defined with reference to the activity of counting doesn't really prevent you from spinning off into wild flights of dsquared-like fancy, because you're pretty soon going to introduce a symbolism that makes things a lot more convenient to express and define, and once you've got the symbolism, it will suggest further things you can do. So while ultimately everything will be grounded in the activity of counting, you can get to such bizarre things as sets of sets of sets pretty fast.
A counting-based reconstruction of, for instance, multiplication is not very hard to define, but it is pretty tedious, and it would be tedious as fuck to define exponentiation that way (not to mention you'd need a lot of things on hand to count!). You'll end up talking about numbers and operations on numbers as a notational convenience even if deep down you really mean complex counting-based operations. And the notation can take on a life of its own.
I really hope the book has a chapter on numerology and the Bible codes.
I mean, I find it a lot more puzzling that the decoded Book of Revelations predicts Obama's election than that mathematics is useful in modeling gravitational attraction or whatever.
I'll take this thread to 1080 all by myself if I have to.
I'll take this thread to 1080 all by myself if I have to.
And that's just one of my strategies.
You only get credit if you can do it without repeating yourself.
254 than that mathematics is useful in modeling gravitational attraction or whatever.
What I was trying to say is that what is mysterious is not that it's "useful in modeling" things, but that some aspects of our world interact in ways that are exactly determined by various (nonelementary) mathematical concepts.
Using d2 rules, there is some sense in which this thread has already exceeded 1080.
Yeah, but who cares about potential infinities.
As usual, I have no idea what point dsquared is trying to convey.
To 216.2, I apologize for my general pissiness in the thread. Here was my problem, which doesn't really justify the pissiness. Much of the early discussion was of the form: "Hey, look at these dogmatic Christian morons writing a know-nothing argument about the philosophy of math based on their prior unexamined beliefs. Here, let me make a dogmatic, know-nothing argument about the philosophy of math based on my prior unexamined beliefs."
I mean, I don't really take any side on the issues argued here and am not competent to do so, but one thing that is clear is that they are not easy issues, and that there are a lot of smart people who have spent a lot of time thinking about them.
This marketing approach definitely does not approaching Christianity or philosophy in good faith. Hence it's open to mockery. I am ready to believe that the authors approached the subject in good faith, but I'd also believe they didn't. Either seems plausible.
denumerable
Ah, that term would have been very handy in my previous (unfortunately tedious) dispute with H-G about infinity.
Hey, look at these dogmatic Christian morons writing a know-nothing argument about the philosophy of math based on their prior unexamined beliefs.
This was actually not my complaint; I have no doubt that the authors of that book are better able to marshal historical arguments (ones with which I disagree, but still) in the service of their point than I am. What annoys me is that way that fundies specifically will marshal intelligent-seeming arguments in favor of the BIG REVEAL that it's actually JESUS! If they were posing philosophical questions about the nature of mathematics in the cause of better understanding the nature of mathematics, that wuold be one thing. But they're posing those (non-trivial!) questions in the service of convincing people that JESUS! It strikes me as a perversion of the intellectual effort that went into developing and arguing those points in the first place.
Which might be unfair, but.
Unrelatedly, I'm not going to go out on a limb and say the arguments in this thread were know-something, but it does seem a bit inaccurate to call them "unexamined"; outside of this thread, where they've been examined plenty, many of the aforementioned commenters have aired their aforementioned beliefs in previous threads on the same topic, and of course several of the contributors seem to have read and engaged with a book or two on the topic.
Here's something from a review that suggests that the book might not be that bad, more like exactly the kind of thing you'd want if you want serious believers to start taking things like math seriously. (n.b., I am not endorsing the book):
Mathematicians have long mused on the fact that beauty seems so akin to truth, on the fact that highly abstract theories could have such robust real-world applications, and on the fact that mathematics appears to be such a universal human phenomenon. It is tempting for many Christians to try using this as a "gotcha" argument for theism. Howell and Bradley present a theistic explanation as a potentially satisfying one, but, they insist, "We want to stop short of suggesting that a naturalistic worldview cannot explain the success of mathematics." They admit the many ways in which mathematics is actually not as effective as our idealized picture suggests; for every abstract theorem that turns out to be a physical law, there are many other results which have nothing to do with reality, or may even be misleading. They also admit that "there are several evolutionary explanations for forms of human cognition, and," they add, "such explanations need not be seen as contradicting Christian beliefs." But, after all, there is still at least something to the notion that Human aesthetic values, and their subsequent use in successful physical theories, dovetail nicely with a Judeo-Christian view that humans are created in the image of God. Whatever being in God's image exactly entails, it seems to include a rational and aesthetic capacity reflective of God's that enables humans to understand and admire his creation. In short, the implications of a Christian worldview offer an attractive explanation for the effectiveness of mathematics.
Once again, faith does not provide the "right answers," but rather helps shapes the questions. As Christian students seek to understand what being made in the image of God might mean in light of modern science, and what modern science might mean in light of faith, it is most helpful to be guided by this even-handed approach.
To 268.2, I do agree that the conversation got a lot better as it went on, and that the characterization in 265 isn't really accurate for the thread as a whole.
Really, though, why is Dick Such sending you this email?
This was the topic of my dissertation.
Only very marginally loosely related: I just yesterday ran across my college admission essays. GOOD GOD. Wow, are those bad. And based on the fact that I was admitted to colleges, I have to assume those were considered to be of at least passably acceptable quality. I'm so glad I'm not on any college admissions committee. I would admit NO ONE.
Sigh... I was going to make a joke about whether the world is just another app running on God's iphone... but it no longer seems appropriate.
Is mathematics discovered or invented
I think it varies... some parts of math seem more or less inevitable (e.g., calculus, differential equations, group theory), and I tend to think that these were "discovered" (usually more than once, by different people). Other parts of math, especially 20th century pure math, seem more a matter of taste, and I tend to think of them as "inventions" (e.g., arithmetic geometry). (However, this could be partly a sign of my own ignorance... people who work on arithmetic geometry certainly seem to think it is natural and elegant. I can say, though, that the barrier to entry for that field is pretty damn high, so people who don't love it generally don't get to the point of understanding it deeply.)