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It's not just science, I can think of an example from pure mathematics. Cantor's work showing the uncountability of the real numbers was derided by prominent me

It's not just science, I can think of an example from pure mathematics. Cantor's work showing the uncountability of the real numbers was derided by prominent members of the mathematical community for decades. This despite the fact that he was able to boil down his argument into a proof so simple it's taught in introductory undergraduate classes. Now, this was at a time when mathematicians were still trying to rigorously define set theory and ground mathematics on a foundation of logic, but still it's a relatively simple and understandable proof. I think the moral of the story is that people are stubborn and are capable of disagreeing about just about everything.

resource0x 6y ago
Bad example. Cantor's theory is controversial to this day. "Classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of Cantor's set theory." -- Hermann Weyl
- openasocket 6y ago
  I wouldn't really consider Weyl as an example of a modern day critic, considering he died in 1955. I think you'd be hard pressed to find a modern mathematician that rejects the diagonalization argument, outside of finitists, which are pretty far from the mainstream. I mean the proof that there is no bijection from a set to its power set is pretty straightforward and constructive. I'm pretty sure the proof is valid even in intuitionist logic.
- cambalache 6y ago
  By whom? Almost all mainstream mathematicians rightfully lionize Cantor's main contributions, as David Hilbert so eloquently put it: "No one shall expel us from the paradise Cantor has created for us" Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können
alisonkisk 6y ago
Cantor proved that R, the set of all reals, cannot be countable. That was never controversial. What's controversial is claiming that R (or any uncountable set) exists, as no one has rigorously proved without adding an axiom.
- openasocket 6y ago
  What axiom do you have to add, are you talking about the axiom of infinity, defining the set of natural numbers? Because you do need that, but without that wouldn't even be able to define uncountability. After that, you can define the reals using Dedekind cuts or Cauchy sequences, which was known at the time (I believe Cantor actually worked on the Cauchy sequence construction). But there's an even simpler uncountable set: the power set of the natural numbers. Super easy to define, and the diagonalization argument falls right out.
  - GoblinSlayer 6y ago
    To construct R you need to repeat Dedekind cut uncountable number of times, i.e. you need to iterate over all real numbers, which would make them countable.
    - openasocket 6y ago
      You don't need any sort of "repetition" of Dedekind cuts to construct the real numbers, it's actually fairly straightforward. The set of all Dedekind cuts is a subset of the power set of rational numbers (i.e. each Dedekind cut is a set of rational numbers) satisfying the following properties: for each Dedekind cut A 1. A is not the empty set 2. A is not the set of all rational numbers 3. A is closed downwards, meaning if x is in A and y < x, then y is in A 4. A has no greatest element, meaning for all x in A, there is a y in A such that y > x. And each Dedekind cut is in one-to-one correspondence with a real number. You can define the usual arithmetic operations on them, show that every rational number has a corresponding Dedekind cut (for any rational q, we have {x in Q | y < q } as the corresponding cut). I haven't seen a proof of the uncountability of Dedekind cuts using the diagonalization argument, but you can prove there is a one-to-one correspondence between Dedekind cuts and Cauchy sequences of rational numbers, which is another construction of the reals. And there is a fairly straightforward proof that those are uncountable using diagonalization. But as you can see, you don't need any sort of repetition to do the construction, it's just a set of sets of rational numbers satisfying a few simple properties. For more information, check out https://en.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...

resource0x 6y ago

Bad example. Cantor's theory is controversial to this day. "Classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of Cantor's set theory." -- Hermann Weyl

openasocket 6y ago
I wouldn't really consider Weyl as an example of a modern day critic, considering he died in 1955. I think you'd be hard pressed to find a modern mathematician that rejects the diagonalization argument, outside of finitists, which are pretty far from the mainstream. I mean the proof that there is no bijection from a set to its power set is pretty straightforward and constructive. I'm pretty sure the proof is valid even in intuitionist logic.
cambalache 6y ago
By whom? Almost all mainstream mathematicians rightfully lionize Cantor's main contributions, as David Hilbert so eloquently put it: "No one shall expel us from the paradise Cantor has created for us" Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können

alisonkisk 6y ago

Cantor proved that R, the set of all reals, cannot be countable. That was never controversial. What's controversial is claiming that R (or any uncountable set) exists, as no one has rigorously proved without adding an axiom.

openasocket 6y ago
What axiom do you have to add, are you talking about the axiom of infinity, defining the set of natural numbers? Because you do need that, but without that wouldn't even be able to define uncountability. After that, you can define the reals using Dedekind cuts or Cauchy sequences, which was known at the time (I believe Cantor actually worked on the Cauchy sequence construction). But there's an even simpler uncountable set: the power set of the natural numbers. Super easy to define, and the diagonalization argument falls right out.
- GoblinSlayer 6y ago
  To construct R you need to repeat Dedekind cut uncountable number of times, i.e. you need to iterate over all real numbers, which would make them countable.
  - openasocket 6y ago
    You don't need any sort of "repetition" of Dedekind cuts to construct the real numbers, it's actually fairly straightforward. The set of all Dedekind cuts is a subset of the power set of rational numbers (i.e. each Dedekind cut is a set of rational numbers) satisfying the following properties: for each Dedekind cut A 1. A is not the empty set 2. A is not the set of all rational numbers 3. A is closed downwards, meaning if x is in A and y < x, then y is in A 4. A has no greatest element, meaning for all x in A, there is a y in A such that y > x. And each Dedekind cut is in one-to-one correspondence with a real number. You can define the usual arithmetic operations on them, show that every rational number has a corresponding Dedekind cut (for any rational q, we have {x in Q | y < q } as the corresponding cut). I haven't seen a proof of the uncountability of Dedekind cuts using the diagonalization argument, but you can prove there is a one-to-one correspondence between Dedekind cuts and Cauchy sequences of rational numbers, which is another construction of the reals. And there is a fairly straightforward proof that those are uncountable using diagonalization. But as you can see, you don't need any sort of repetition to do the construction, it's just a set of sets of rational numbers satisfying a few simple properties. For more information, check out https://en.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...