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> these algebraic demonstrations are not rigorous proofs because we have not shown that elementary rules for addition and multiplication extend to repeating dec

> these algebraic demonstrations are not rigorous proofs because we have not shown that elementary rules for addition and multiplication extend to repeating decimal numbers too. They do but we have not demonstrated that here. That's why I don't like proofs. There are many things that are not demonstrated here. The same can also be said about the other proof that is presumably complete. Unless you build everything from axioms up, any proof will be incomplete. And building from axioms is too verbose. From the practical perspective it's like proving that 1 is equal to 1.0000...

r0uv3n 4y ago
Building from axioms is entirely possible, and in fact you'll often see the construction of first the rational numbers from the natural numbers and then the real numbers from Cauchy sequences or Dirichlet cuts within the first few weeks of studying mathematics in university. The construction of the natural numbers itself is often not discussed deeply, but the actual details aren't all that hard.
- sobriquet9 4y ago
  Everything you know is easy for you. For others, there's opportunity cost. The time one spent on learning Peano axioms is the time one could have spent on learning something more useful, say, linear algebra.
  - xigoi 4y ago
    You can take all the well-known properties of numbers as axioms and study linear algebra. Then later, if you're curious, you can prove the presumptions from a smaller set of axioms.
Someone 4y ago
> That's why I don't like proofs. If you don’t like them, I think grudgingly accepting them as a necessary evil is your best option. If you don’t do that, what do you use instead? Intuition? Beliefs? Do you take any statement that, when expressed from the axioms up, is too complex, as unknown?
- sobriquet9 4y ago
  I look at math the same way as at physics, where rigorous proofs are not possible. The difference only matters to the few people who do it professionally. I know I can dig deeper all the way down to axioms if I choose so, but realistically there is not enough time for me to learn all of the underlying math. At most, I can occasionally check if the results are consistent and match reality.
  - jfengel 4y ago
    It seems like the kind of thing that's not really worth having an opinion on. As you say, the difference only amounts to a few people -- but for those few people, they can draw the distinction in a rigorous fashion. If you're trying to push it all the way to epistemology, then even the most trivial proof doesn't really work. "A, A->B, therefore B" could be wrong because you have only your brain to inspect it. And computers, and other logicians, but they could all be glitchy. Even "Assume A, therefore A" could be wrong. This is a valid line of reasoning, but if one wants to follow it, it would seem more useful to apply it to trivial proofs rather than get bogged down in a problem people have a weirdly difficult time accepting. I'd kinda like somebody to investigate the properties of proofs that non-mathematicians feel compelled to have an opinion in, though.
  - constantcrying 4y ago
    >where rigorous proofs are not possible. Absurd. >I know I can dig deeper all the way down to axioms if I choose so, but realistically there is not enough time for me to learn all of the underlying math. So proofs are impossible because you refuse to learn math? What.
    - sobriquet9 4y ago
      > Absurd. Show me the proof of Newton's laws of motion.
rthnbgrredf 4y ago
I suggest to just you to think of proofs in a different way. There are different ways to write a proof. I think proofs are sufficient if you are able to convince your audience. If you try to convince a theorem prover like Coq you have to be very explicit of all the annoying details otherwise Coq will complain. It's very tedious work and some of the reasons why it's not more commonly used among programmers. If you write a math paper you can assume that your audience is quite knowledgeable in the domain and you don't have to be very explicit about every detail (readers can fill in the gaps) but you have to be precise in your notation and reasoning. You provide a formal proof that convinces your peers not a rigid machine. If you write a blog post then informal proofs are very much sufficient, e.g. 0.3333... = 1/3 0.9999... = 3/3 = 1 Is sufficient to convince your audience. You don't have to write down a fully scientific formal proof or Coq program to elaborate your argument in the context of a blog post.
- sobriquet9 4y ago
  The problem with this formulation is that one can convince the audience that something is right when it isn't. Example: 31 is prime, 331 is prime, 3331 is prime, 33331 is prime, ... 33333331 is prime, therefore 333333331 must be prime (while in fact it's divisible by 17). I'm not arguing one should never look at proofs and take everything for granted. I'm saying that rigorous proofs going down all the way to Peano axioms are unnecessary. In fact, most math people actually use, including linear algebra, calculus, Fourier transform predates Peano axioms.
  - nh23423fefe 4y ago
    I don't see the analogy. you asserted a finite set of premises but didn't prove the implication at all? so why would the conclusion be valid?
    - sobriquet9 4y ago
      My point is that if all I need is to convince my audience, I may be able to make them believe the flawed proof. In this example the flaw is intentionally obvious, but there are other examples where the flaw is more subtle.
constantcrying 4y ago
>And building from axioms is too verbose. Ridicolous. You use previous results which were built up from axioms. The correct formal proof here is very short, it is just an appeal to the definition of equality for two real numbers.

r0uv3n 4y ago

Building from axioms is entirely possible, and in fact you'll often see the construction of first the rational numbers from the natural numbers and then the real numbers from Cauchy sequences or Dirichlet cuts within the first few weeks of studying mathematics in university. The construction of the natural numbers itself is often not discussed deeply, but the actual details aren't all that hard.

sobriquet9 4y ago
Everything you know is easy for you. For others, there's opportunity cost. The time one spent on learning Peano axioms is the time one could have spent on learning something more useful, say, linear algebra.
- xigoi 4y ago
  You can take all the well-known properties of numbers as axioms and study linear algebra. Then later, if you're curious, you can prove the presumptions from a smaller set of axioms.

Someone 4y ago

> That's why I don't like proofs. If you don’t like them, I think grudgingly accepting them as a necessary evil is your best option. If you don’t do that, what do you use instead? Intuition? Beliefs? Do you take any statement that, when expressed from the axioms up, is too complex, as unknown?

sobriquet9 4y ago
I look at math the same way as at physics, where rigorous proofs are not possible. The difference only matters to the few people who do it professionally. I know I can dig deeper all the way down to axioms if I choose so, but realistically there is not enough time for me to learn all of the underlying math. At most, I can occasionally check if the results are consistent and match reality.
- jfengel 4y ago
  It seems like the kind of thing that's not really worth having an opinion on. As you say, the difference only amounts to a few people -- but for those few people, they can draw the distinction in a rigorous fashion. If you're trying to push it all the way to epistemology, then even the most trivial proof doesn't really work. "A, A->B, therefore B" could be wrong because you have only your brain to inspect it. And computers, and other logicians, but they could all be glitchy. Even "Assume A, therefore A" could be wrong. This is a valid line of reasoning, but if one wants to follow it, it would seem more useful to apply it to trivial proofs rather than get bogged down in a problem people have a weirdly difficult time accepting. I'd kinda like somebody to investigate the properties of proofs that non-mathematicians feel compelled to have an opinion in, though.
- constantcrying 4y ago
  >where rigorous proofs are not possible. Absurd. >I know I can dig deeper all the way down to axioms if I choose so, but realistically there is not enough time for me to learn all of the underlying math. So proofs are impossible because you refuse to learn math? What.
  - sobriquet9 4y ago
    > Absurd. Show me the proof of Newton's laws of motion.

rthnbgrredf 4y ago

I suggest to just you to think of proofs in a different way. There are different ways to write a proof. I think proofs are sufficient if you are able to convince your audience. If you try to convince a theorem prover like Coq you have to be very explicit of all the annoying details otherwise Coq will complain. It's very tedious work and some of the reasons why it's not more commonly used among programmers. If you write a math paper you can assume that your audience is quite knowledgeable in the domain and you don't have to be very explicit about every detail (readers can fill in the gaps) but you have to be precise in your notation and reasoning. You provide a formal proof that convinces your peers not a rigid machine. If you write a blog post then informal proofs are very much sufficient, e.g. 0.3333... = 1/3 0.9999... = 3/3 = 1 Is sufficient to convince your audience. You don't have to write down a fully scientific formal proof or Coq program to elaborate your argument in the context of a blog post.

sobriquet9 4y ago
The problem with this formulation is that one can convince the audience that something is right when it isn't. Example: 31 is prime, 331 is prime, 3331 is prime, 33331 is prime, ... 33333331 is prime, therefore 333333331 must be prime (while in fact it's divisible by 17). I'm not arguing one should never look at proofs and take everything for granted. I'm saying that rigorous proofs going down all the way to Peano axioms are unnecessary. In fact, most math people actually use, including linear algebra, calculus, Fourier transform predates Peano axioms.
- nh23423fefe 4y ago
  I don't see the analogy. you asserted a finite set of premises but didn't prove the implication at all? so why would the conclusion be valid?
  - sobriquet9 4y ago
    My point is that if all I need is to convince my audience, I may be able to make them believe the flawed proof. In this example the flaw is intentionally obvious, but there are other examples where the flaw is more subtle.

constantcrying 4y ago

>And building from axioms is too verbose. Ridicolous. You use previous results which were built up from axioms. The correct formal proof here is very short, it is just an appeal to the definition of equality for two real numbers.