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The biggest problem in mathematics has never been proving theorems. Mathematics was never focused on proof until the formalization of math became popular approx

The biggest problem in mathematics has never been proving theorems. Mathematics was never focused on proof until the formalization of math became popular approximately 100 years ago. Mathematics is used as a tool to explain the world and to create the least-complex model which predicts the data. So computers can aid us in verifying our work--but ultimately we aren't interested in determining all the possible logical theorems. Instead, we're only interested in finding the logically valid theorems which serve the above purpose of explaining the patterns in the world concisely. It's a subtle difference, but for computers to solve that problem requires for them to have a certain understanding of humans and our goals with science. I think that's more than 10 years out.

Jimmy 7y ago
>Instead, we're only interested in finding the logically valid theorems which serve the above purpose of explaining the patterns in the world concisely. That's just empirically false though. Take the "perfectoid spaces" mentioned in the article. Do you think mathematicians are interested in them because they "explain the patterns in the world concisely"? No, they're interested in them because of their mathematical significance, and their applications to problems within mathematics. Mathematicians are not physicists.
- solinent 7y ago
  > mathematical significance, and their applications to problems within mathematics These external problems ultimately are about explaining patterns in the world. I'm just saying you'll never replace the relationship between the mathematician and the physicist by simply replacing the mathematician. Instead, you'd have to replace the whole academic process, making fully autonomous universities. In which case they're still not interested in exactly same problems unless they are constructed in the same way (which they are not, since they are not human). I'm not saying that computers won't be able to aid us with these meta-theories which help push our main theories forward. So I do agree that they can come up with some theory, but not all of it, and not most of it.
baby 7y ago
What? Number Theory was a field of study that saw literally 0 applications up until cryptography became a serious field of study.
- mjcohen 7y ago
  The number theoretician Edmund Landau said that number theory was a very practical application for getting a Ph.D. in mathematics.
ncmncm 7y ago
Mathematicians choose what to keep according to aesthetics. Mathematics can only be defined as what mathematicians like. For a long time, they didn't like irrationals, so anything involving those didn't count as mathematics. Zero took a long time to be accepted, starting in India. Negative solutions of quadratics were illegitimate until astonishingly recent days. Complex numbers were accepted even more recently. Greek geometers knew they lived on a sphere, but spherical geometry was too unpleasant to contemplate until quite recently, when it turned out interesting theorems could live there. People worked outside these boundaries all along, but what they wrote didn't catch on. Laplace's transforms were ignored and forgotten until they were needed to shore up Heaviside's extremely practical D operator. Complex numbers turned out to be needed to for electromagnetics. Once people got deeply into the topic, they discovered beauty and then mathematics accepted them. Mathematics is the world's largest and longest-running effort to produce a collective work of sublime beauty. What is beautiful goes in, what isn't dies with its creator. New forms come to be seen as beautiful as they are shown to open new vistas to explore, but very slowly.
- chrisweekly 7y ago
  Interesting, and artfully put. Are you borrowing / paraphrasing, or is this your "original" idea?
  - ncmncm 7y ago
    Might be original. Just an observation, really.
    - chrisweekly 7y ago
      An astute one. Kudos!
- jacobolus 7y ago
  > For a long time, they didn't like irrationals, so anything involving those didn't count as mathematics. Note that this is more or less a myth. > Greek geometers knew they lived on a sphere, but spherical geometry was too unpleasant to contemplate until quite recently Astronomers did a huge amount of sophisticated spherical geometry, from Mesopotamians through e.g. Hipparchus and later Ptolemy, then Arabs/Persians, Indians, medieval Europeans, right down to the present.
nikofeyn 7y ago
> Mathematics is used as a tool to explain the world and to create the least-complex model which predicts the data. that's maybe a description of how mathematics is used, but that isn't a description of mathematics.
keiru 7y ago
>Mathematics was never focused on proof until the formalization of math became popular approximately 100 years ago. We are transitioning surprisingly fast from kinda explaining stuff to automatically and systematically advancing exclusively through rigurously formalized proofs.

Jimmy 7y ago

>Instead, we're only interested in finding the logically valid theorems which serve the above purpose of explaining the patterns in the world concisely. That's just empirically false though. Take the "perfectoid spaces" mentioned in the article. Do you think mathematicians are interested in them because they "explain the patterns in the world concisely"? No, they're interested in them because of their mathematical significance, and their applications to problems within mathematics. Mathematicians are not physicists.

solinent 7y ago
> mathematical significance, and their applications to problems within mathematics These external problems ultimately are about explaining patterns in the world. I'm just saying you'll never replace the relationship between the mathematician and the physicist by simply replacing the mathematician. Instead, you'd have to replace the whole academic process, making fully autonomous universities. In which case they're still not interested in exactly same problems unless they are constructed in the same way (which they are not, since they are not human). I'm not saying that computers won't be able to aid us with these meta-theories which help push our main theories forward. So I do agree that they can come up with some theory, but not all of it, and not most of it.

baby 7y ago

What? Number Theory was a field of study that saw literally 0 applications up until cryptography became a serious field of study.

mjcohen 7y ago
The number theoretician Edmund Landau said that number theory was a very practical application for getting a Ph.D. in mathematics.

ncmncm 7y ago

Mathematicians choose what to keep according to aesthetics. Mathematics can only be defined as what mathematicians like. For a long time, they didn't like irrationals, so anything involving those didn't count as mathematics. Zero took a long time to be accepted, starting in India. Negative solutions of quadratics were illegitimate until astonishingly recent days. Complex numbers were accepted even more recently. Greek geometers knew they lived on a sphere, but spherical geometry was too unpleasant to contemplate until quite recently, when it turned out interesting theorems could live there. People worked outside these boundaries all along, but what they wrote didn't catch on. Laplace's transforms were ignored and forgotten until they were needed to shore up Heaviside's extremely practical D operator. Complex numbers turned out to be needed to for electromagnetics. Once people got deeply into the topic, they discovered beauty and then mathematics accepted them. Mathematics is the world's largest and longest-running effort to produce a collective work of sublime beauty. What is beautiful goes in, what isn't dies with its creator. New forms come to be seen as beautiful as they are shown to open new vistas to explore, but very slowly.

chrisweekly 7y ago
Interesting, and artfully put. Are you borrowing / paraphrasing, or is this your "original" idea?
- ncmncm 7y ago
  Might be original. Just an observation, really.
  - chrisweekly 7y ago
    An astute one. Kudos!
jacobolus 7y ago
> For a long time, they didn't like irrationals, so anything involving those didn't count as mathematics. Note that this is more or less a myth. > Greek geometers knew they lived on a sphere, but spherical geometry was too unpleasant to contemplate until quite recently Astronomers did a huge amount of sophisticated spherical geometry, from Mesopotamians through e.g. Hipparchus and later Ptolemy, then Arabs/Persians, Indians, medieval Europeans, right down to the present.

nikofeyn 7y ago

> Mathematics is used as a tool to explain the world and to create the least-complex model which predicts the data. that's maybe a description of how mathematics is used, but that isn't a description of mathematics.

keiru 7y ago

>Mathematics was never focused on proof until the formalization of math became popular approximately 100 years ago. We are transitioning surprisingly fast from kinda explaining stuff to automatically and systematically advancing exclusively through rigurously formalized proofs.