51–60 of 282 posts
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My understanding is that since any mathematical proof can (in principle) be reduced to manipulation with formal objects (symbols), it’s always “constructive.”
Not really. Check out the proof of Hilbert's Nullstellensatz. He got tons of criticism for proving the existence of a relation without constructing that relation.
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From a genetic standpoint, all humans are 99.9% identical beasts. There is of course some genetic components in potential abilities. But I strongly believe that most of the differences we are seeing are built on compounded interest of environmental factors, starting from a very young age.
I'm not sure this is a viable argument considering that we also share much of our DNA with apes/monkeys. It seems like minor genetic differences can create large disparities. I definitely think environmental factors can compound on a genetic baseline though, and of course the earlier you start compounding the greater the returns.
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> there will be a sensible way to define the zero/empty case. This is often a good test of whether you have a uniform concept that works for all n without corner cases. And so, begun the array indexing war has.
A polynomial always includes a member (monomial) with the power zero. It seems natural, therefore, to index the coefficients correspondingly. In other situations, 1 may be the more natural starting index.
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From a genetic standpoint, all humans are 99.9% identical beasts. There is of course some genetic components in potential abilities. But I strongly believe that most of the differences we are seeing are built on compounded interest of environmental factors, starting from a very young age.
Zero chance it’s environmental
I often use von Neumann's quote, "Young man, in mathematics you don't understand things. You just get used to them.", to console myself when I have to read a math book multiple times to really understand something. The quote also has truth in it. I had no problem accepting that 0! = 1 only because I learnt that fact early in school. However, I struggled quite a bit to accept that span({}) = {0}, even though it is not…
0! = 1 is not hard to accept, since it follows a rule. You just need to look at it backwards. To get the previous factorial (Ni-1)! you need to divide the N! by N 4! = 24 ;; 24 / 4 = 6 3! = 6 ;; 6 / 3 = 2 2! = 2 ;; 2 / 2 = 1 1!= 1 ;; 1 / 1 = 1 0! = 1
Maybe these things take time, but it just doesn't seem like there is anything remotely like this going on today, despite far greater access to education and increases in global population.
Is it perhaps that the limelight is less on mathematicians than it was during the early 20th century?
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Yes, those two facts about zero/empty cases (and so many more) are definitely related, and this class of facts is one of my favourites! Usually, if you're dealing with something algebraic in flavour (which is a very vague concept, sorry), there will be a sensible way to define the zero/empty case. This is often a good test of whether you have a uniform concept that works for all n without corner cases. It almost irri…
Yeah, it took a while to sink into my head that many of these "wait, why is span({}) = {0}?" kinds of cases have answers that sum up as "because anything else means other rules are inconsistent, and the whole thing is either less useful or useless". It's "arbitrary", but it's either the only useful option, or sometimes a simple(st) one of many. Even just one number theory course helped a lot, since it brought that ki…
Is it just me or does it seem like there something like there were a lot of celebrity-level STEM heroes to come out of the early 20th century. These folks all seemed to work together closely. Maybe these things take time, but it just doesn't seem like there is anything remotely like this going on today, despite far greater access to education and increases in global population. Is it perhaps that the limelight is les…
Is it just me or does it seem like there something like there were a lot of celebrity-level STEM heroes to come out of the early 20th century. These folks all seemed to work together closely. Maybe these things take time, but it just doesn't seem like there is anything remotely like this going on today, despite far greater access to education and increases in global population. Is it perhaps that the limelight is les…
The fact that so many of them specifically grew up in Hungary, went to school in Germany, and came to the United States is interesting to note though!
I often use von Neumann's quote, "Young man, in mathematics you don't understand things. You just get used to them.", to console myself when I have to read a math book multiple times to really understand something. The quote also has truth in it. I had no problem accepting that 0! = 1 only because I learnt that fact early in school. However, I struggled quite a bit to accept that span({}) = {0}, even though it is not…
I wonder if e.g. using constructive math would make things easier since every step of a constructive proof is understandable.