4 ms·

The mention of the Russell-Whitehead proof brought back memories. It's from Principia Mathematica (same title as Newton's book, but it's a different work.) Here

The mention of the Russell-Whitehead proof brought back memories. It's from Principia Mathematica (same title as Newton's book, but it's a different work.) Here's a representative page: http://quod.lib.umich.edu/cache/a/a/t/aat3201.0001.001/00000401.tifs.gif http://quod.lib.umich.edu/cache/a/a/t/aat3201.0001.001/00000... Look at the last line of 54:43. Then look at the page number. This isn't a joke. It's really how bad axiomatic mathematics is. During my last year of high school and first year of college, I was fascinated by this stuff to the point that I couldn't think about anything else. After digesting Peano arithmetic, I moved on to Zermelo-Frenkel set theory (the "set theory" that we normally use is self-contradictory and therefore meaningless to a mathematician.) I worked through Godel's incompleteness theorem. I tried my best to understand non-Euclidean geometries. And a bunch of other stuff that I don't even care to remember now. Soon, predictably, it started affecting my health. At that point, I broke down and gave up that shit for good. The stuff I do today is positively quotidian by comparison. But it makes me happy. Even now, I'm occasionally uneasy about living in a universe where we don't know (actually, can't know) if the continuum hypothesis is true or false. But I shove the thought quickly from my head. The point of all this is to tell you guys that unless you really know what you're getting into, don't think too deeply about the meaning of proof :-)

jerf 18y ago
In our actual, factual physical universe, the Continuum Hypothesis isn't so much "true" or "false" as "irrelevant". It's possible the universe is fundamentally discrete, in which case it just doesn't apply to anything physical, and even if it turns out to be continuous in some sense, it's likely to be some hybrid continuous/discrete combo where the hypothesis still doesn't apply to anything physical. (One of the points made in Reflections on Relativity ( http://www.mathpages.com/rr/rrtoc.htm http://www.mathpages.com/rr/rrtoc.htm ) is that both a continuous universe and a discrete universe, taking the traditional senses of the term, are logically contradictory things to build a universe out of, and Einsteinian relativity calls for an odd mixture of both. No clean link to a single page as, IIRC, the point is made over a series of sections, culminating in a unique (AFAIK) discussion of Zeno's Paradoxes, which I can link to: http://www.mathpages.com/rr/s3-07/3-07.htm http://www.mathpages.com/rr/s3-07/3-07.htm although you really need to read the sections before that for full impact.) So, sleep easy. It is nothing more and nothing less than a choice of axiom. The axiom of choice is similarly irrelevant in the real universe; since use of the axiom calls for the selection of an infinite set from another infinite set, possibly uncountable, it has no particular connection to the real, finite universe.
- randomwalker 18y ago
  That makes sense, and it's likely that if I read through the entire book that you linked to, I'd grasp it pretty clearly. But it was certainly out of my mental reach as an 18-year old. And believe me, I tried. In general, it appears that our ability to ask philosophical questions, especially of the mathematical variety, far outpaces our ability to find answers or even comprehend answers that others have come up with. It is highly plausible that this is linked to the disturbingly high occurrence of depression, bordering on mental illness, among mathematicians. (I know enough mathematicians personally for that observation to have some statistical validity, but anyone who has read many biographies of mathematicians should get the sense of it.) What I'm trying to say is that while many of the questions of mathematical philosophy pique our curiosity greatly, and it can certainly be a rewarding experience to get a taste of the beauty of the field, everyone should have a mental threshold for how much they are willing to get involved. Perhaps this is obvious to other people, but since it was a painful lesson for me, I feel obliged to share it :-)
ars 18y ago
I thought the reason it was so complicated is that he used symbolic logic to do it - maybe that's not the right way to go about these proofs? It seemed more like a work in showing the use of symbolic logic, rather than a proof of math.
rw 18y ago
Mathematical Platonism is dangerous for mental health. Cantor is a great example of someone who literally went crazy contemplating the meaning of uncountable infinites. Set theorists, by the way, seem to be the most likely to go insane.
- antiform 18y ago
  What are you basing this on? Mathematics may lead to highly surprising and nonintuitive answers, but that doesn't mean that mathematical ideas drive people crazy. While Cantor did take his work seriously and was said to have believed that he was indeed doing God's work, he was probably bipolar. Given the attitude toward mental health at the time, it seems that whether he was a mathematician, merchant, or mason, he would have become "mad." While Cantor is the classic example of a mathematician gone insane, there was also a number of things besides grappling with concepts of infinity that could have also have been a factor, like always being passed over for professorships and many prominent mathematicians (like Henri Poincare) trying to keep his work out of the canon of mathematics.
  - rw 18y ago
    Thank you for adding relevant biographical details.

jerf 18y ago

In our actual, factual physical universe, the Continuum Hypothesis isn't so much "true" or "false" as "irrelevant". It's possible the universe is fundamentally discrete, in which case it just doesn't apply to anything physical, and even if it turns out to be continuous in some sense, it's likely to be some hybrid continuous/discrete combo where the hypothesis still doesn't apply to anything physical. (One of the points made in Reflections on Relativity ( http://www.mathpages.com/rr/rrtoc.htm http://www.mathpages.com/rr/rrtoc.htm ) is that both a continuous universe and a discrete universe, taking the traditional senses of the term, are logically contradictory things to build a universe out of, and Einsteinian relativity calls for an odd mixture of both. No clean link to a single page as, IIRC, the point is made over a series of sections, culminating in a unique (AFAIK) discussion of Zeno's Paradoxes, which I can link to: http://www.mathpages.com/rr/s3-07/3-07.htm http://www.mathpages.com/rr/s3-07/3-07.htm although you really need to read the sections before that for full impact.) So, sleep easy. It is nothing more and nothing less than a choice of axiom. The axiom of choice is similarly irrelevant in the real universe; since use of the axiom calls for the selection of an infinite set from another infinite set, possibly uncountable, it has no particular connection to the real, finite universe.

randomwalker 18y ago
That makes sense, and it's likely that if I read through the entire book that you linked to, I'd grasp it pretty clearly. But it was certainly out of my mental reach as an 18-year old. And believe me, I tried. In general, it appears that our ability to ask philosophical questions, especially of the mathematical variety, far outpaces our ability to find answers or even comprehend answers that others have come up with. It is highly plausible that this is linked to the disturbingly high occurrence of depression, bordering on mental illness, among mathematicians. (I know enough mathematicians personally for that observation to have some statistical validity, but anyone who has read many biographies of mathematicians should get the sense of it.) What I'm trying to say is that while many of the questions of mathematical philosophy pique our curiosity greatly, and it can certainly be a rewarding experience to get a taste of the beauty of the field, everyone should have a mental threshold for how much they are willing to get involved. Perhaps this is obvious to other people, but since it was a painful lesson for me, I feel obliged to share it :-)

ars 18y ago

I thought the reason it was so complicated is that he used symbolic logic to do it - maybe that's not the right way to go about these proofs? It seemed more like a work in showing the use of symbolic logic, rather than a proof of math.

rw 18y ago

Mathematical Platonism is dangerous for mental health. Cantor is a great example of someone who literally went crazy contemplating the meaning of uncountable infinites. Set theorists, by the way, seem to be the most likely to go insane.

antiform 18y ago
What are you basing this on? Mathematics may lead to highly surprising and nonintuitive answers, but that doesn't mean that mathematical ideas drive people crazy. While Cantor did take his work seriously and was said to have believed that he was indeed doing God's work, he was probably bipolar. Given the attitude toward mental health at the time, it seems that whether he was a mathematician, merchant, or mason, he would have become "mad." While Cantor is the classic example of a mathematician gone insane, there was also a number of things besides grappling with concepts of infinity that could have also have been a factor, like always being passed over for professorships and many prominent mathematicians (like Henri Poincare) trying to keep his work out of the canon of mathematics.
- rw 18y ago
  Thank you for adding relevant biographical details.