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>Its central dogma is thou should prove everything rigorously. >To me, that's the entire point. You prove things rigorously so that (among many other reasons)

by ldp01 9y ago

>Its central dogma is thou should prove everything rigorously. >To me, that's the entire point. You prove things rigorously so that (among many other reasons) when someone considers trying to square the circle, you can point them to the proof that the circle cannot be squared. This. Those defending mathematical rigour are a minority almost everywhere except a school of mathematics. There is no shortage of people trying all the slap-dash ways of doing things.

protonfish 9y ago
Then why do I frequently get confronted by the religion of "inherent truth in math" and we keep teaching our children math made overly convoluted and unintuitive by the "rigorous" crowd? Pretending that this is only a minority is not helpful in quashing the damage done by magical thinking in mathematics. An example in computer science is the whole P = NP nonsense. We have never found any evidence that P = NP, nor has anyone discovered a compelling reason why it should be. Yet, the first sentence on its Wikipedia page is: > The P versus NP problem is a major unsolved problem in computer science. "Unsolved?" Only in the perspective of the "rigorous proof" crowd which you imply are an unimportant minority. The truth is that P = NP is just another, as Dr. Zeilberger calls it, "stupid question."
- Ceezy 9y ago
  This problem IS actually unsolved. Maybe you found something very smart to tell us that nobody knows...?
  - protonfish 9y ago
    It depends what you mean by "solved." In a rigorous, pure math sense, no. But so what? Math is only about itself and its own esoteric rules, not about practical reality. In a scientific sense, it is proven in the same way we know there are no unicorns: there isn't a scrap of evidence they exist, nor is there any reason to believe they should.
    - ldp01 9y ago
      We use math as a framework for reasoning about physical reality. When doing this we assume that the logic of our math holds for physical reality and that we can use it to make useful predictions. If a mathematical theory has held up so-far then that is evidence that it will continue to. It's not a guarantee, but it is evidence.
- nessus42 9y ago
  I don't get your P = NP example. If P = NP, it has serious consequences. E.g., as I understand it, a lot of cryptography will ultimately fail if we discover that P = NP. We have no reason to believe that P = NP, but on the other hand, without proof, we can't know for certain that it doesn't. We're not infinitely smart, and we just might not have yet seen the way in which P does equal NP. Surely we live in a world filled with uncertainty, and we have to be able to deal with that. But in instances in which we can be certain about certain things, it certainly benefits us. Also, I don't understand the argument that learning to think with the rigor of mathematics is somehow detrimental. There are many different ways of reasoning, and a well-educated person should strive to learn a good amount of these useful mental tools. Learning to think with mathematical rigor is a good tool to have in one's mental toolbox.
  - wolfgke 9y ago
    > a lot of cryptography will ultimately fail if we discover that P = NP. This is conjectured, but can you prove it? When the degree of the polynomial of the minimal running time or its coefficients gets large, one can imagine a world where e.g. asymmetric cryptography, as we understand it today, it still possible.
- ocb 9y ago
  P = NP is interesting precisely because of what you're saying: it feels trivially obvious that they aren't equal but no one can actually prove it... which sort of implies that it's not trivially obvious.
- ldp01 9y ago
  > Then why do I frequently get confronted by the religion of "inherent truth in math" and we keep teaching our children math made overly convoluted and unintuitive by the "rigorous" crowd? Pretending that this is only a minority is not helpful in quashing the damage done by magical thinking in mathematics. Are you able to expand on this point? In my experience I have not heard of children learning rigorous mathematics. Grade school kids learn arithmetic. High school kids learn some calculus and probability (at most). This curriculum is practical for those going to uni or a technical trade. (although the way it's taught is critical to be of any use. If I were a science teacher my class would just build arduino projects all day!) Your reference to magical thinking seems to be the opposite of what I would consider rigour... I would associate magical thinking with an economist or engineer building a sprawling quantitative risk model and then justifying billion dollar decisions with it. Later realising a small logical error has invalidated their conclusion or that they have overfit their data etc. (Just for example...)

by ldp01 9y ago

protonfish 9y ago

Then why do I frequently get confronted by the religion of "inherent truth in math" and we keep teaching our children math made overly convoluted and unintuitive by the "rigorous" crowd? Pretending that this is only a minority is not helpful in quashing the damage done by magical thinking in mathematics. An example in computer science is the whole P = NP nonsense. We have never found any evidence that P = NP, nor has anyone discovered a compelling reason why it should be. Yet, the first sentence on its Wikipedia page is: > The P versus NP problem is a major unsolved problem in computer science. "Unsolved?" Only in the perspective of the "rigorous proof" crowd which you imply are an unimportant minority. The truth is that P = NP is just another, as Dr. Zeilberger calls it, "stupid question."

Ceezy 9y ago
This problem IS actually unsolved. Maybe you found something very smart to tell us that nobody knows...?
- protonfish 9y ago
  It depends what you mean by "solved." In a rigorous, pure math sense, no. But so what? Math is only about itself and its own esoteric rules, not about practical reality. In a scientific sense, it is proven in the same way we know there are no unicorns: there isn't a scrap of evidence they exist, nor is there any reason to believe they should.
  - ldp01 9y ago
    We use math as a framework for reasoning about physical reality. When doing this we assume that the logic of our math holds for physical reality and that we can use it to make useful predictions. If a mathematical theory has held up so-far then that is evidence that it will continue to. It's not a guarantee, but it is evidence.
nessus42 9y ago
I don't get your P = NP example. If P = NP, it has serious consequences. E.g., as I understand it, a lot of cryptography will ultimately fail if we discover that P = NP. We have no reason to believe that P = NP, but on the other hand, without proof, we can't know for certain that it doesn't. We're not infinitely smart, and we just might not have yet seen the way in which P does equal NP. Surely we live in a world filled with uncertainty, and we have to be able to deal with that. But in instances in which we can be certain about certain things, it certainly benefits us. Also, I don't understand the argument that learning to think with the rigor of mathematics is somehow detrimental. There are many different ways of reasoning, and a well-educated person should strive to learn a good amount of these useful mental tools. Learning to think with mathematical rigor is a good tool to have in one's mental toolbox.
- wolfgke 9y ago
  > a lot of cryptography will ultimately fail if we discover that P = NP. This is conjectured, but can you prove it? When the degree of the polynomial of the minimal running time or its coefficients gets large, one can imagine a world where e.g. asymmetric cryptography, as we understand it today, it still possible.
ocb 9y ago
P = NP is interesting precisely because of what you're saying: it feels trivially obvious that they aren't equal but no one can actually prove it... which sort of implies that it's not trivially obvious.
ldp01 9y ago
> Then why do I frequently get confronted by the religion of "inherent truth in math" and we keep teaching our children math made overly convoluted and unintuitive by the "rigorous" crowd? Pretending that this is only a minority is not helpful in quashing the damage done by magical thinking in mathematics. Are you able to expand on this point? In my experience I have not heard of children learning rigorous mathematics. Grade school kids learn arithmetic. High school kids learn some calculus and probability (at most). This curriculum is practical for those going to uni or a technical trade. (although the way it's taught is critical to be of any use. If I were a science teacher my class would just build arduino projects all day!) Your reference to magical thinking seems to be the opposite of what I would consider rigour... I would associate magical thinking with an economist or engineer building a sprawling quantitative risk model and then justifying billion dollar decisions with it. Later realising a small logical error has invalidated their conclusion or that they have overfit their data etc. (Just for example...)