4 ms·

Is there something mysterious about mathematics?

72 points by fbrusch 11y ago

andrewstuart2 11y ago
So, kind of a summary of/riff on last week's Nova? Then again, maybe just a coincidence. http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html
ahelwer 11y ago
I think we can all relate to an unclear sense of awe upon first encountering something like e^(i * pi) = -1 (which might have occurred around the first time reading Contact). It takes exposure to a certain amount of math to switch paradigms: it would be weirder if that identity weren't the case. Dazzling inexperienced students with magical identities and coincidences isn't teaching mathematics, it's teaching numerology. I really appreciated this article.
- gphil 11y ago
  Yeah, once you work through the explanation of that kind of thing the magic kind of falls out of it when you realize how the definitions fit together to produce that result. However, I think there are certainly some areas of Math that are still pretty mysterious to the human intellect, no matter how advanced. The distribution of prime numbers and chaos come to mind.
  - tel 11y ago
    The Monster group is my favorite example. It may very well have the same nature of "just falling out of definitions in an unsurprising manner" but it's certainly too large for any human to ever feel that way. Monstrous Moonshine was this project and is mentioned in the article.
- xamuel 11y ago
  In an ideal world, anyone trying to dazzle students with e^(i*pi) should immediately be rebuffed with the demand: "define complex exponentiation". While I was teaching calculus, I found that students have a tendency to assume that any function sufficiently familiar to them, must be defined for whatever values you can think to stick in it; if you were teaching the mathematics of chairs and tables, they wouldn't bat an eye if you mentioned cos(chair). This phenomenon also explains why there's such a ridiculously big deal made by laymen about the undefinedness of division-by-zero.
  - impendia 11y ago
    On the one hand, I agree with you. On the other hand, to answer your demand: I define complex exponentiation by the formula e^(it) = -1 for all real, nonzero t. Clearly stupid. Why? There are a lot of properties that one might naturally ask e^(it) to satisfy (for example, that e^(it) * e^(iu) = e^(i(t + u)), or that its derivative is ie^(it)), and the usual complex exponential satisfies all the right ones. So writing e^(i * pi) with no context has, I submit, at least some content.
platz 11y ago
I wonder what opinion this author would have of constructivist mathematics
- wetmore 11y ago
  Why is that?
schoen 11y ago
Gregory Chaitin, the developer of algorithmic information theory, has repeatedly said that many or almost all mathematical facts are "true for no reason". https://en.wikipedia.org/wiki/Gregory_Chaitin#Other_scholarly_contributions https://en.wikipedia.org/wiki/Gregory_Chaitin#Other_scholarl... A simple way of trying to understanding Chaitin's view is that if you try to take mathematical facts and match them up with explanations or proofs that humans could understand or recognize as useful or elegant, most facts won't be able to be matched up with any such explanation, because the facts inherently outnumber the explanations, even in a set-theoretic sense. But it might be better to take a look at Chaitin's explanation rather than my paraphrase.
- vegedor 11y ago
  If it's pure math, sure, like string theory in theoretical physics, but a lot of maths is well founded and needed for application. That drives the direction of research a lot. "true for no reason" is oxymoronic
- fsk 11y ago
  It sounds like he's talking about the incompleteness theorem, that there are statements that are true but we can never find a proof. That's closely related to the halting problem and the idea of computability. There are some programs that will run forever without halting, but we have no method for identifying all of them. ("true and not provable" matches "runs forever without halting, but we can't prove it")
tel 11y ago
I really enjoy Saunders MacLane's Mathematics Form and Function for sight at the "answers" this article seeks. MacLane's argument is that mathematics arises from a relatively small number of meaningful forces and "good" mathematics arises when those forces all align in particular places. You could of course go further by asking what exactly it is that makes those forces all align—why in these fields?—but the point of his book isn't exactly to answer the philosophical question but more to provide a little bit of a humanistic POV on the development of mathematics with just enough historical happenstance stripped away to make things intelligible but not so much as to lose the line of sight on why things (probably) developed the way that they did.
calhoun137 11y ago
Physics is about the way things are, whereas math is about the way things have to be.
danbruc 11y ago
I am not really surprised that there are patterns and connections between seemingly unrelated concepts. Why? Because I think they are way less unrelated than it may seem. After all everything is build on top of a pretty small number of axioms. Every result in number theory - proven or yet unproven - is essentially a consequence of the Peano axioms and the definition of the operations on those numbers. The Riemann hypothesis is a statement about prime numbers, which numbers are prime is defined by the Peano axioms and the definition of multiplication. The real numbers are in some sense build on top of the natural numbers by going through the rational numbers. And you can build the natural numbers on top of set theory. So in essence I think we are just exploring the structure of one and the same object - or maybe a few objects - and every branch of mathematics does this by making a few more assumptions and therefore looking only on a part of the whole.
ankurdhama 11y ago
No.

72 points by fbrusch 11y ago

So, kind of a summary of/riff on last week's Nova? Then again, maybe just a coincidence. http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html

ahelwer 11y ago

I think we can all relate to an unclear sense of awe upon first encountering something like e^(i * pi) = -1 (which might have occurred around the first time reading Contact). It takes exposure to a certain amount of math to switch paradigms: it would be weirder if that identity weren't the case. Dazzling inexperienced students with magical identities and coincidences isn't teaching mathematics, it's teaching numerology. I really appreciated this article.

gphil 11y ago
Yeah, once you work through the explanation of that kind of thing the magic kind of falls out of it when you realize how the definitions fit together to produce that result. However, I think there are certainly some areas of Math that are still pretty mysterious to the human intellect, no matter how advanced. The distribution of prime numbers and chaos come to mind.
- tel 11y ago
  The Monster group is my favorite example. It may very well have the same nature of "just falling out of definitions in an unsurprising manner" but it's certainly too large for any human to ever feel that way. Monstrous Moonshine was this project and is mentioned in the article.
xamuel 11y ago
In an ideal world, anyone trying to dazzle students with e^(i*pi) should immediately be rebuffed with the demand: "define complex exponentiation". While I was teaching calculus, I found that students have a tendency to assume that any function sufficiently familiar to them, must be defined for whatever values you can think to stick in it; if you were teaching the mathematics of chairs and tables, they wouldn't bat an eye if you mentioned cos(chair). This phenomenon also explains why there's such a ridiculously big deal made by laymen about the undefinedness of division-by-zero.
- impendia 11y ago
  On the one hand, I agree with you. On the other hand, to answer your demand: I define complex exponentiation by the formula e^(it) = -1 for all real, nonzero t. Clearly stupid. Why? There are a lot of properties that one might naturally ask e^(it) to satisfy (for example, that e^(it) * e^(iu) = e^(i(t + u)), or that its derivative is ie^(it)), and the usual complex exponential satisfies all the right ones. So writing e^(i * pi) with no context has, I submit, at least some content.

platz 11y ago

I wonder what opinion this author would have of constructivist mathematics

wetmore 11y ago
Why is that?

schoen 11y ago

Gregory Chaitin, the developer of algorithmic information theory, has repeatedly said that many or almost all mathematical facts are "true for no reason". https://en.wikipedia.org/wiki/Gregory_Chaitin#Other_scholarly_contributions https://en.wikipedia.org/wiki/Gregory_Chaitin#Other_scholarl... A simple way of trying to understanding Chaitin's view is that if you try to take mathematical facts and match them up with explanations or proofs that humans could understand or recognize as useful or elegant, most facts won't be able to be matched up with any such explanation, because the facts inherently outnumber the explanations, even in a set-theoretic sense. But it might be better to take a look at Chaitin's explanation rather than my paraphrase.

vegedor 11y ago
If it's pure math, sure, like string theory in theoretical physics, but a lot of maths is well founded and needed for application. That drives the direction of research a lot. "true for no reason" is oxymoronic
fsk 11y ago
It sounds like he's talking about the incompleteness theorem, that there are statements that are true but we can never find a proof. That's closely related to the halting problem and the idea of computability. There are some programs that will run forever without halting, but we have no method for identifying all of them. ("true and not provable" matches "runs forever without halting, but we can't prove it")

tel 11y ago

I really enjoy Saunders MacLane's Mathematics Form and Function for sight at the "answers" this article seeks. MacLane's argument is that mathematics arises from a relatively small number of meaningful forces and "good" mathematics arises when those forces all align in particular places. You could of course go further by asking what exactly it is that makes those forces all align—why in these fields?—but the point of his book isn't exactly to answer the philosophical question but more to provide a little bit of a humanistic POV on the development of mathematics with just enough historical happenstance stripped away to make things intelligible but not so much as to lose the line of sight on why things (probably) developed the way that they did.

calhoun137 11y ago

Physics is about the way things are, whereas math is about the way things have to be.

danbruc 11y ago

I am not really surprised that there are patterns and connections between seemingly unrelated concepts. Why? Because I think they are way less unrelated than it may seem. After all everything is build on top of a pretty small number of axioms. Every result in number theory - proven or yet unproven - is essentially a consequence of the Peano axioms and the definition of the operations on those numbers. The Riemann hypothesis is a statement about prime numbers, which numbers are prime is defined by the Peano axioms and the definition of multiplication. The real numbers are in some sense build on top of the natural numbers by going through the rational numbers. And you can build the natural numbers on top of set theory. So in essence I think we are just exploring the structure of one and the same object - or maybe a few objects - and every branch of mathematics does this by making a few more assumptions and therefore looking only on a part of the whole.

ankurdhama 11y ago

No.